GIFT   OF 
MICHAEL  REESE 


THE 


ELASTICITY  AND  RESISTANCE 


OF   THE 


MATERIALS    OF    ENGINEERING, 


BY 


WM.  H.  BURR,  C.E., 

WILLIAM    HOWARD    HART    PROFESSOR   OF   RATIONAL  AND   TECHNICAL 
MECHANICS   AT   RENSSELAER   POLYTECHNIC   INSTITUTE. 


UNIVERSITY 


NEW  YORK : 

JOHN   WILEY   &   SONS. 
1883. 


b 


COPYRIGHT,  1883, 
BY  WM.  H.  BURR. 


PRESS   OF    4.    J.    LITTLE   i  CO., 


PREFACE. 


THIS  work  has  been  the  outgrowth  of  lectures  on  the 
elasticity  and  resistance  of  materials,  given  by  the  author  to 
succeeding  classes  of  students  in  the  department  of  civil  engi- 
neering at  the  Rensselaer  Polytechnic  Institute.  Although 
those  lectures,  as  given,  form  the  basis  of  the  work,  they  have, 
of  course,  been  considerably  elaborated  and  extended,  so  as  to 
cover  many  details  of  the  subject  which  it  would  be  impossible 
to  include  in  any  ordinary  technical  course  of  study,  but  which, 
at  the  same  time,  are  necessary  to  a  complete  and  philosoph- 
ical treatment. 

The  first,  or  "  Rational,"  part  of  this  work,  is  intended  to 
furnish  an  analytical  or  rational  basis  for  the  "  Technical"  or 
practical  development  contained  in  Part  II.  It  will  undoubt- 
edly impress  a  great  number,  and  perhaps  all  engineers  in 
active  practice,  that  it  is  unnecessary  to  the  proper  treatment 
of  such  a  subject.  Indeed,  a  very  considerably  extended 
experience  in  iron  and  steel  constructions  places  the  author 
himself  in  position  to  fully  appreciate  the  weight  of  such  a 
criticism  at  the  first  glance.  But  it  may  be  contended,  and  he 
thinks  must  be  admitted,  that  the  present  advanced  state  of 
engineering  as  a  profession  implies  the  existence  of  something 
that  may  be  called  the  "  natural  philosophy  "  of  engineering. 
In  other  words,  the  engineer  of  the  present  time  must  meet 
the  increased  and  increasing  demands  upon  him  in  some 
one  or  more  specialty,  not  only  by  the  aid  of  sound  common 


IV  PREFACE. 


sense  and  a  well-trained  judgment,  but  also  by  a  systematic 
knowledge  of  so  much  of  natural  philosophy  as  is  involved 
in  practical  engineering  operations.  The  ideal  simplicity  of 
stresses  and  strains  in  a  perfectly  isotropic  body,  and  the  clear- 
ness of  action  of  "  external  forces  "  applied  at  any  "  point  "  or 
distributed  over  any  "  surface  "  according  to  some  known  and 
well-defined  law,  are  not,  it  is  evident,  the  things  the  technical 
student  will  encounter  in  his  practice  as  an  engineer.  He  will 
find  few,  if  any,  of  the  ideal  conditions  realized,  and  the  diffi- 
culties constantly  confronting  him  will  be  those  involving 
modifications  of  the  analytical  or  mathematical  results  based 
upon  ideal  quantities  and  conditions.  Nevertheless,  it  is  cer- 
tainly true  that  in  engineering  practice  he  deals  with  precisely 
the  same  quantities  as  in  the  natural  philosophy  of  engineering, 
but  in  different  amounts  and  with  far  different  and  vastly  more 
complicated  conditions.  And  it  is  equally  true  that  a  correct 
knowledge  of  the  consequent  mpdifications,  both  in  kind  and 
amount  must  be  based  not  only  upon  a  correct  recognition  of 
the  actual  circumstances  into  which  the  ideal  conditions  trans- 
mute themselves  in  engineering  works,  *>.,  upon  sound  prac- 
tical knowledge,  but  also  upon  a  thorough  comprehension  of 
the  things  involved,  in  the  abstract,  and  the  laws  governing 
their  actions  'and  relations.  In  other  words,  but  in  essentials 
the  same,  an  engineer's  preparation  for  active  practice  must 
consist  both  of  that  philosophical  training  in  what  is  largely 
ideal,  and  which  he  acquires  in  the  technical  school,  and  of  the 
purely  practical  training  of  the  first  few  years  of  his  profes- 
sional life. 

The  first,  or  "  Rational,"  part  of  this  work  is,  then,  designed 
for  few  others  than  technical  students,  although  there  are 
engineers  whose  tastes  induce  or  circumstances  require  inves- 
tigations in  connection  with  the  elasticity  and  resistance  of 
materials.  The  writer  would  esteem  himself  fortunate  if  the 
mathematical  portion  of  the  book  should  find  favor  with  such 
individuals  and  be  useful  to  them. 


PREFACE. 


In  Part  II.  the  mathematical  results  obtained  in  Part  I.  are 
subjected  to  the  test  of  experiment.  By  the  aid  of  experi- 
mental results  in  a  great  variety  of  material,  empirical  coeffi- 
cients are  established  which  involve  the  varied  and  complicated 
circumstances  of  material  in  actual  use.  The  formulae,  which 
otherwise  express  ideal  conditions  only,  are  thus  rendered  of 
the  greatest  practical  value  ;  in  fact  they  constitute  the  only 
reliable  practical  formulae  in  use  by  engineers. 

All  the  experimental  results  are,  of  course,  compilations 
only,  but  they  have  been  taken  in  all  cases  from  what  are 
believed  to  be  trustworthy  sources,  and  it  has  been  the  inten- 
tion to  give  credit  to  the  experimenter  in  every  case.  It  may 
appear  that  too  great  a  profusion  of  experimental  results  has 
been  introduced.  But  it  has  been  the  aim  of  the  author,  even 
at  the  risk  of  being  tedious,  to  represent  truly  and  completely 
the  great  variety  of  both  quantitative  and  qualitative  phenom- 
ena exhibited  by  material  un^ier  test  ;  to  show  not  only  the 
variation  in  products  of  different  mills  but  the  variation  in 
different  products  of  the  same  mill ;  to  exhibit  the  variations 
due  to  difference  in  size,  shape,  relative  dimensions  and  condi- 
tion*of  specimens  ;  to  show  that  specimens  apparently  identi- 
cally the  same  may  even  give  considerable  diversity  in  results 
and  to  prove  the  difference  between  the  finished  member  and 
its  component  parts,  as  well  as  to  indicate  the  direction  in 
which  further  investigations  may  most  profitably  be  prose- 
cuted. A  few  groups  of  tests  are  not  sufficient  to  the  attain- 
ment of  such  a  series  of  results. 

In  the  course  of  the  preparation  of  the  MSS.  the  author 
found  it  necessary  to  reduce  a  very  great  amount  of  experi- 
mental quantities  from  the  crude  shape  of  a  mere  record  of 
tests  to  a  useful  condition,  and  to  change  many  others  from 
one  unit  to  another.  These  numerical  operations  involved 
much  labor,  and  although  they  were  performed  with  great  care 
and  repeated  in  almost  every  instance,  it  is  very  probable  that 
errors  have  crept  in,  though  it  is  believed  that  there  are  few, 


VI  PREFACE. 


if  any,  of  importance.  The  writer  will  feel  indebted  to  any 
one  who  will  discover  them.  In  all  cases,  unless  otherwise 
specifically  stated,  the  ultimate  resistance,  elastic  limit  and 
coefficient  of  elasticity  are  expressed  in  pounds  per  square 
inch  of  original  area  of  section. 

In  a  few  of  the  tables  of  Art.  32  the  "strains,"  i.e.t  amounts 
of  stretch,  are  given  as  decimal  fractions  (hundredths)  of  orig- 
inal length,  while  the  otherwise  uniform  method  of  expression 
is  by  means  of  whole  numbers  giving  per  cents,  of  original 
dimensions.  This  diversity  is  unintentional  and  due  to  the 
fact  that  a  part  of  the  MSS.  was  a  portion  of  that  used  in 
lectures. 

The  distinction  between  "  stress  "  and  "  strain  "  conflicts, 
so  far  as  the  latter  word  is  concerned,  with  ordinary  usage. 
But  some  distinction  is  absolutely  necessary,  and  that  used  has 
had  a  long  existence,  and  is  at  least  consistent  with  the  ety- 
mology of  the  words.  There  certainly  can  be  no  way  of 
filling  the  hiatus  caused  by  the  absence  of  a  word  to  concisely 
express  changes  of  shape  or  dimensions,  without  some  incon- 
venience, and  that  followed  will  probably  cause  as  little  as 
any. 

W.  H.  B. 

RENSSELAER  POLYTECHNIC  INSTITUTE, 

1883. 


CONTENTS. 


PART  I.— RATIONAL. 
CHAPTER  I. 

GENERAL    THEORY    OF    ELASTICITY    IN    AMORPHOUS    SOLID 

BODIES. 

ART.  PACE 

I. — General  Statements I 

2. — Coefficients  of  Elasticity 3 

3. — Lateral  Strains 4 

\. — Relation  between  the  Coefficients  of  Elasticity  for  Shearing  and  Direct 

Stress  in  a  Homogeneous  Body 6 

J, — Expressions  for  Tangential  and  Direct  Stresses  in  Terms  of  the  Rates  of 

Strains  at  any  Point  of  a  Homogeneous  Body 8 

6 — General  Equations  of  Internal  Motion  and  Equilibrium 14 

7 — Equations  of  Motion  and  Equilibrium  in  Semi-Polar  Co-ordinates 2O 

8. — Equations  of  Motion  and  Equilibrium  in  Polar  Co-ordinates 27 

CHAPTER  .II. 

THICK,   HOLLOW   CYLINDERS  AND   SPHERES,  AND  TORSION. 

9. -Thick,  Hollow  Cylinders 36 

10.— Torsion  in  Equilibrium 43 

Equations  of  condition  in  rectangular  co-ordinates 5° 

Solutions  of  Equations  (13)  and  (21) 53 

Elliptical  section  about  its  centre 54 

Equilateral  triangle  about  its  centre  of  gravity 57 


Vlll  CONTENTS. 


ART.  PAGE 

4  Rectangular  section  about  an  axis  passing  through  its  centre  of 

gravity 59 

Circular  section  about  its  centre 75 

General  observations 77 

ii. — Torsional  Oscillations  of  Circular  Cylinders 78 

12.— Thick,  Hollow  Spheres 84 


CHAPTER   III. 

THE    ENERGY    OF    ELASTICITY. 

13. — Work  Expended  in  Producing  Strains 90 

14. — Resilience 96 

15. — Suddenly  Applied  External  Forces  or  Loads 98 

16. — Longitudinal  Oscillations  of  a  Straight  Bar  of  Uniform  Section 100 


CHAPTER  IV. 

THEORY    OF    FLEXURE. 

17. — General  Formulae i( 

18. — The  Common  Theory  of  Flexure 122 

19. — Deflection  by  the  Common  Theory  of  Flexure no 

20. — External  Bending  Moments  and  Shears  in  General    1^2 

21. — Moments  and  Shears  in  Special  Cases IJ8 

Case  I '. ij8 

Case  II 139 

Case  III #2 

22. — Recapitulation   of  theV  General    Formulae    of   the    Common    Theory   of 

Flexure :43 

23.— The  Theorem  of  Three  Moments r46 

230. — Reactions  under  Continuous  Beam  of  any  Number  of  Spans *  57 

Special  method  for  ordinary  use ^65 

Special  case  of  equal  spans 169 

24. — The  Neutral  Curve  for  Special  Cases 


Cascl. 


171 
172 

Case  IT JI74 

Case  III 177 


CONTENTS.  ix 


ART. 

25. — The  Flexure  of  Long  Columns 

26. — Graphical  Determination  of  the  Resistance  of  a  Beam !o(j 

27.— The  Common  Theory  of  Flexure  with  Unequal  Values  of  Coefficients  of 

Elasticity i99 

28. — Greatest  Stresses  at  any  Point  in  a  Beam 2O2 


PART  II.— TECHNICAL. 
CHAPTER  V. 

TENSION. 

29. — General  Observations. — Limit  of  Elasticity 208 

30. — Ultimate  Resistance 210 

31. — Ductility. — Permanent  Set 211 

32. — Wrought  Iron 212 

Coefficient  of  elasticity 212 

Ultimate  resistance  and  elastic  limit 223 

Wrought-iron  boiler  plate 241 

Effect  of  annealing 245 

Effect  of  hardening  on  the  tensile  resistance  of  iron  and  steel. . .  247 

Variation  of  tensile  resistance  luith  increase  of  temperature . ...  247 

Effect  of  low  temperatures  on  'wrought  iron 251 

Iron  "wire 254 

Tensile  resistance  of  shape  iron 257 

English  wrought  iron 258 

Fracture  of  wrougJii  iron 258 

Crystallization  of  "wrought  iron 259 

Elevation  of  ultimate  resistance  and  elastic  limit 260 

Bauschingcr's  experiments  on  the  change  of  elastic  limit  and 

coefficient 2o2 

Resistance  of  bar  iron  to  suddenly  applied  stress 269 

Reduction  of  resistance  between  the  ultimate  and  breaking  points.  269 

Effects  of  chemical  constitution 27(> 

Kirkaldy's  conclusions 272 

33.— Cast  Iron 2 

Coefficient  of  elasticity  and  elastic  limit 27& 

Ultimate  resistance 279 

Effect  of  rcmelting 28a 


CONTENTS. 


ART.  PAGE 

Effect  of  continued  fusion 284 

Effect  of  repetition  of  stress 284 

Effect  of  high  temperatures 286 

34.— Steel 286 

Coefficient  of  elasticity 286 

Ultimate  resistance  and  elastic  limit 294 

Boiler  plate 306 

Effects  of  hardening  and  tempering  steel  plates 314 

Rivet  steel 315 

Effect  of  reduction  of  sectional  area,  in  connection  with  ham- 
mering and  rolling 315 

Effects  of  annealing  steel 319 

Steel  wire 319 

Shape  steel 321 

Steel  gun  wire 322 

Effect  of  low  and  high  temperatures  on  steel ,   323 

Effect  of  manipulations  common  to  constructive  processes  ;  also 

punched^  drilled  and  reamed  holes 325 

Bauschinger1  s  experiments  on  the  change  of  elastic  limit  and 

coefficient  of  elasticity 333 

Fracture  of  steel 334 

Effect  of  chemical  composition 334 

35. — Copper,  Tin,  Zinc  and  their  Alloys. — Phosphor  Bronze 336 

Coefficients  of  elasticity 336 

Ultimate  resistance  and  elastic  limit 338 

Gun  metal 341 

General  results 342 

Phosphor  bronze,  and  brass  and  copper  wire 344 

Experiments  on  rolled  copper  by  the  "Franklin  Institute  Com- 
mittee " 345 

Variation  of  ultimate  resistance  and  stretch  at  high  tempera- 
tures    345 

Bauschinger1  s  experiments  with  copper  and  red  brass 348 

36. — Various  Metals  and  Glass 351 

Coefficients  of  elasticity 351 

Ultimate  resistance  and  elastic  limit 352 

37. — Cement,  Cement  Mortars,  etc. — Brick 353 

Experiments  and  conclusions  of  Wm.  W.  Maclay,  C.  E 357 

Artificial  stones 363 

Bricks 364 

Adhesion  between  bricks  and  cement  mortar 364 

38.— Timber 365 


CONTENTS.  XJ 


CHAPTER  VI. 

COMPRESSION. 

ART. 

39.—  Preliminary  ............................................. 

40.—  Wrought  Iron  ..........................................  ' 

41.  —  Cast  Iron  ................................. 

'*" 


...~~ 

43.  —  Copper,  Tin,  Zinc  and  their  Alloys  ...........................  -35 

44.-Glass  ..........................................  ...."/.•.".".".!."."..".".!  389 

45.  —  Cement.  —  Cement  Mortar.  —  Concrete.  —  Artificial  Stones  ...............   300 

46.—  Brick  ......................................................  '    ""  3Q6 

47.  —  Natural  Building  Stones  .................................. 

48.—  Timber  .....................................  .....!.......!.....!!  403 


CHAPTER    VII. 
COMPRESSION. — LONG   COLUMNS. 

49- — Preliminary  Matter 409 

5°- — Gordon's  Formula ; 430 

51. — Experiments  on  Phoenix  Columns,  Latticed  Channel  Columns  and  Chan- 

nels • ^^Q 

Latticed  channel  columns  and  channels 455 

52. — Euler's  and  Tredgold's  Forms  of  Long  Column  Formula 463 

53- — Hodgkinson's  Formulas ^69 

54- — Graphical  Representation  of  Results  of  Long  Column  Experiments 473 

55- — Limit  of  Applicability  of  Euler's  Formula 477 

56. — Reduction  of  Columns  at  Ends 479 

57« — Timber  Columns 480 

Formula  of  C.  Shaler  Smith,  C.  E 485 


CHAPTER  VIII. 

SHEARING  AND   TORSION. 

58.— Coefficient  of  Elasticity 4§7 

59.— Ultimate  Resistance 49° 

Wrought  iron -4  'r 

Cast  iron. .  4'  <3 


xil  CONTENTS. 


ART.  PAGE 

Steel 493 

Copper 495 

Timber 496 

60. — Torsion 498 

Coefficients  of  elasticity 498 

Ultimate  resistance  and  elastic  limit 498 

Wrought  iron 498 

Cast  iron 500 

Steel 504 

Copper,  tin,  zinc  and  their  alloys 507 

Timber 509 

Relation  between  ultimate  resistances  to  tension  and  torsion 510 


CHAPTER  IX. 

BENDING,    OR   FLEXURE. 

61. — Coefficient  of  Elasticity 512 

62. — Formulae  for  Rupture 512 

63. — Solid  Rectangular  and  Circular  Beams 514 

Wrought  iron 515 

Cast  iron 518 

Steel 520 

Combined  steel  and  iron 523 

Copper,  tin,  zinc  and  their  alloys 524 

Timber  beams 5 26 

Timber  beams  of  natural  and  prepared  "wood 536 

Cement,  mortar  and  concrete 537 

Stone  beams 543 

Practical  formula  for  solid  beams 543 

Comparison  of  modulus  of  rupture  for  bending  with  ultimate 

resistances 545 

64. — Flanged  Beams  with  Unequal  Flanges 546 

Equal  coefficients  of  elasticity '. 547 

Unequal  coefficients  of  elasticity 550 

Cast-iron  flanged  beams 5  54 

Deflection  of  cast-iron  flanged  beams 55^ 

Wrought-iron  T  beams 559 

JDeflection  of  wrought-iron  T  beams 5^2 

65. — Flanged  Beams  with  Equal  Flanges 564 

Experiments  of  U.  S.  Test  Board 575 


CONTENTS. 


66.— Built  Flanged  Beams  with  Equal  Flanges.— Cover  Plates 57g 

67.— Built  Flanged  Beams  with  Equal  Flanges.— No  Cover  Plates 

68.— Box  Beams " 

69. — Exact  Formulae  for  Built  Beams co6 

70.— Examples  of  Built  Beams  Broken  by  Centre  Weight 598 

Example  I. —  Wrought-iron  beam 598 

Example  II. — Steel  beam .  500 

71. — Loss  of  Metal  at  Rivet  Holes 50! 

72.— Thickness  of  Web  Plate 


CHAPTER  X. 

CONNECTIONS. 

73.— Riveted  Joints 606 

Kinds  of  joints 606 

Distribution  of  stress  in  riveted  joints 607 

Effect  of  punching 615 

Wrought-iron  lap  joints ,  and  butt  joints  'with  single  butt  strap.  616 

Steel  lap  joints,  and  butt  joints  -with  one  cover 627 

Wrought-iron  butt  joints  with  double  covers 632 

Steel  butt  joints  -with  double  cover  plates 635 

Efficiencies 638 

Riveted  truss  joints 640 

Diagonal  joints 642 

Friction  of  riveted  joints 642 

Hand  and  machine  riveting 643 

74. — Welded  Joints 643 

75. — Pin  Connection ''44 

76.— Iron,  Steel  and  Hemp  Cables  or  Ropes.— Wrought-iron  Chain  Cables. . .  648 

Wrought-iron  chain  cables 652 


CHAPTER  XI. 

MISCELLANEOUS    PROBLEMS. 

77.— Resistance  of  Flues  to  Collapse 6 

78.— Approximate  Treatment  of  Solid  Metallic  Rollers ° 

79.— Resistance  to  Driving  and  Drawing  Spikes 6 

80.— Shearing  Resistance  of  Timber  behind  Bolt  or  Mortise  Holes 664 

81.— Bulging  of  Plates ^ 


xiv  CONTENTS. 

ART.  PAGE 

82. — Special  Cases  of  Flexure 674 

Flexure  by  oblique  forces 674 

General  flexure  by  continuous,  normal  load 679 

CHAPTER  XII. 

WORKING  STRESSES   AND    SAFETY   FACTORS. 

83. — Definitions 681 

84. — Specifications  for  Sabula  Bridge 682 

85. — Specifications  for  Albany  and  Greenbush  Bridge 686 

86. — Niagara  Suspension  Bridge 688 

87. — Menomonee  Draw  Bridge 689 

88. — Franklin  Square  Bridge 694 

89. — General  Specifications 699 

90. — New  York,  Chicago  and  St.  Louis  Railway  Specifications 699 

91.— Plattsmouth  Bridge 701 

92. — Specifications  for  Steel  Cable  Wire  for  the  East  River  Suspension  Bridge.  703 
93. — Specifications  for  Steel  Wire  Ropes  for  the  Over-Floor  Stays  and  Storm 

Cables  of  the  East  River  Suspension  Bridge 705 

94. — Specifications  for  Steel  Suspenders,  Connecting  Rods,  Stirrups  and  Pins, 

for  the  East  River  Suspension  Bridge 705 

95. — Specifications  for  Certain  Steel  Work  .  .  .  East  River  Bridge, 

1881 706 


CHAPTER  XIII. 

THE    FATIGUE    OF    METALS. 

96. — Wohler's  Law 708 

97. — Experimental  Results 709 

98. — Formulae  of  Launhardt  and  Weyrauch 715 

99. — Influence  of  Time  on  Strains 719 

CHAPTER   XIV. 

THE    FLOW    OF    SOLIDS. 

100. — General  Statements 723 

101. — Tresca's  Hypotheses 725 


CONTENTS.  xv 


ART.  PAGE 

102. — The  Variable  Meridian  Section  of  the  Primitive  Central  Cylinder 727 

103. — Positions  in  the  Jet  of  Horizontal  Sections  of  the  Primitive  Central 

Cylinder 729 

104. — Final  Radius  of  a  Horizontal  Section  of  the  Primitive  Central  Cylinder.  731 

105. — Path  of  any  Molecule 731 


ADDENDA. 

To  Art.  20 734 

To  Art.  73 737 

To  Art.  75 737 


UNIVERSITY 


ELASTICITY  AND  RESISTANCE  OF 
MATERIALS. 


PART  I.— RATIONAL. 


CHAPTER   I. 

GENERAL  THEORY  OF  ELASTICITY  IN  AMORPHOUS  SOLID 

BODIES. 


Art.  i.— General   Statements. 

THE  molecules  of  all  solid  bodies  known  in  nature  are  more 
or  less  free  to  move  toward,  or  from,  or  among  each  other. 
Resistances  are  offered  to  such  motions,  which  vary  according 
to  the  circumstances  under  which  they  take  place,  and  the 
nature  of  the  body.  This  property  of  resistance  is  termed  the 
"  elasticity  "  of  the  body. 

The  summation  of  the  displacements  of  the  molecules  of  a 
body,  for  a  given  point,  is  called  the  "  distortion"  or  "strain  " 
at  the  point  considered.  The  force  by  which  the  molecules  of 
a  body  resist  a  strain,  at  any  point,  is  called  the  "stress  "  at 
that  point.  This  distinction  between  stress  and  strain  is  fun- 
damental  and  important. 

Stresses  are  developed,  and  strains  caused,  by  the  applica- 
tion of  force  to  the  exterior  surface  of  the  material.  These 
stresses  and  strains  vary  in  character  according  to  the  method 


.  \ 

.  ;  ;      :   / 

2  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  I. 

of  application  of  the  external  forces.  Each  stress,  however,  is 
accompanied  by  its  own  characteristic  strain  and  no  other. 
Thus,  there  are  shearing  stresses  and  shearing  strains,  tensile 
stresses  and  tensile  strains,  compressive  stresses  and  compres- 
sive  strains.  Usually  a  number  of  different  stresses  with  their 
corresponding  strains  are  coexistent  at  any  point  in  a  body 
subjected  to  the  action  of  external  forces. 

It  is  a  matter  of  experience  that  strains  always  vary  con- 
tinuously and  in  the  same  direction  with  the  corresponding 
stresses.  Consequently  the  stresses  are  continuously  increasing 
functions  of  the  strains,  and  any  stress  may  be  represented 
by  a  series  composed  of  the  ascending  powers  (commencing 
with  the  first)  of  the  strains  multiplied  by  proper  coefficients. 
When,  as  is  usually  the  case,  the  displacements  are  very  small, 
the  terms  of  the  series  whose  indices  are  greater  than  unity 
are  exceedingly  small  compared  with  the  first  term,  whose 
index  is  unity.  Those  terms  may  consequently  be  omitted 
without  essentially  changing  the  value  of  the  expression. 
Hence  follows  what  is  ordinarily  termed  Hooke's  law  : 

The  ratio  between  stresses  and  corresponding  strains,  for  a 
given  material,  is  constant. 

This  law  is  susceptible  of  very  simple  algebraic  representa- 
tion. As  the  generality  of  the  equation  will  not  be  affected, 
intensities  of  stresses  and  distortions  or  strains  per  linear  unit, 
only,  will  be  considered. 

Let/'  represent  the  intensity  of  any  stress,  and  /'  the  strain 
per  unit  of  length,  or,  in  other  words,  the  rate  of  strain.  If 
E'  is  a  constant  coefficient,  Hooke's  law  will  be  given  by  the 
following  equation  : 

...   .....    (') 


If  the  intensity  of  stress  varies  from  point  to  point  of  a  body, 
Hooke's  law  may  be  expressed  by  the  following  differential 
equation  : 


-  2'1  COEFFICIENTS  OF  ELASTICITY. 


dl1 


If/  and  /'  are  rectangular  co-ordinates,  Eqs.  (i)  and  (2)  are 
evidently  the  equations  of  a  straight  line  passing  through  the 
origin  of  co-ordinates.  It  will  hereafter  be  seen  that  the  line 
under  consideration  is  essentially  straight  for  very  small.strains 
only. 

Art.  2.— Coefficients  of  Elasticity. 

In  general,  the  coefficient  E'  in  Eq.  (i)  of  the  preceding 
Art.,  is  called  the  "  coefficient  of  elasticity,"  or,  sometimes, 
"modulus  of  elasticity."  -The  coefficient  of  elasticity  varies 
both  with  the  kind  of  material  and  kind  of  stress.  It  simply 
expresses  the  ratio  between  stress  and  strain. 

The  characteristic  strain  of  a  tensile  stress  is  evidently  an 
increase  of  the  linear  dimensions  of  the  body  in  the  direction 
of  action  of  the  external  forces. 

Let  this  increase  per  unit  of  length  be  represented  by  /, 
while/  and  E  represent,  respectively,  the  corresponding  in- 
tensity and  coefficient.  Eq.  (i)  of  the  preceding  Art.  then 
becomes : 


=  El,     or,     £  = 


E  is  then  the  coefficient  of  elasticity  for  tension. 

The  characteristic  strain  for  a  compressive  stress  is  evi- 
dently a  decrease  in  the  linear  dimensions  of  the  body  in  the 
direction  of  action  of  the  external  forces.  Let  7X  represent  this 
decrease  per  unit  of  length,  /t  the  intensity  of  compressive 
stress,  and  El  the  corresponding  coefficient.  Hence : 


or,        ,== 


ELASTICITY  IN  AMORPHOUS   SOLID  BODIES.     [Art.  3. 
f,,  consequently,  is  the  coefficient  of  elasticity  for  compres- 


sion. 

The  characteristic  strain  for  a  shearing  stress  may  be  deter- 
mined by  considering  the  effect  which  it  produces  on  the  layers 
of  the  body  parallel  to  its  plane  of  action. 

In  Fig.  i  let  ABCD  represent  one  face  of  a  cube,  another 
of  whose  faces  is  fixed  along  AD.     If  a  shear  acts  in  the  face 
BC,  whose  plane  is  normal  to  the  plane  of  the 
paper,   all  layers    of  the    cube  parallel   to   the 
plane  of  the  shearing  stress,  i.e.,  BC,  will  slide 
over  each  other,  so  that  the  faces  AB  and  DC 
will    take   the   positions  AE   and   DF.      The 
amount  of  distortion  or  strain  per  unit  of  length 
will  be  represented  by  the  angle  EAB  =  cp.  -If 
r  the  strain  is  small  there  may  be  written  cp,  sin  cp 

or  tan  cp  indifferently. 
Representing,  therefore,  the  intensity  of  shear,  coefficient 
and  strain  by  S,  G  and  cp,  respectively,  Eq.  (i)  of  Art.  i  bq- 
comes : 


Fig.1 


5  =  Gcp,     or,      G  — 


(3) 


It  will  be  seen  hereafter  that  there  are  certain  limits  of 
stress  within  which  Eqs.  (i),  (2)  and  (3)  are  essentially  true, 
but  beyond  which  they  do  not  hold  ;  this  limit  is  called  the 
"  limit  of  elasticity,"  and  is  not  in  general  a  well  defined 
point. 


Art.  3. — Lateral   Strains. 

If  a  body,  like  that  shown  in  Fig.  i,  be  subjected  to  ten- 
sion, all  of  its  oblique  cross  sections,  such  as  FE  and  GH,  will 
sustain  shearing  stresses  in  consequence  of  the  components 
of  the  tension  tangential  to  those  oblique  sections.  These 


Art.  3.]  LATERAL   STRAINS.  5 

tangential  stresses  will  cause  the  oblique  sections,  in  both 
directions,  to  slide  over  each  other.  Consequently  the  normal 
cross  sections  of  the  body  will  be  decreased ;  and  if  the  normal 


V\\NI-  x'''X'' 

\>XX/>' 

J1I 

./*^ 

3                 F                    H                         C 

Fig.1 

> 

cross  sections  of  the  body  are  made  less,  its  capacity  of  resist- 
ance to  the  external  forces  acting  on  AB  and  CD  will  be  cor- 
respondingly diminished. 

If  the  body  is  subjected  to  compression,  oblique  sections  of 
the  body  will  be  subjected  to  shears,  but  in  directions  opposite 
to  those  existing  in  the  previous  case.  The  effect  of  such 
shears  will  be  an  increase  of  the  lateral  dimensions  of  the  body 
and  a  corresponding  increase  in  its  capacity  of  resistance. 

These  changes  in  the  lateral  dimensions  of  the  body  are 
termed"7'  lateral  strains  ";  they  always  accompany  direct  strains 
of  tension  arrd  compression. 

It  is  to  be  observed  that  lateral  strains  decrease  a  body's 
resistance  to  tension,  but  increase  its  resistance  to  compression. 
Also,  that  if  they  are  prevented,  both  kinds  of  resistance  are 
increased. 

Consider  a  cube,  each  of  whose  edges  is  a,  in  a  body  sub- 
jected to  tension.  Let  r  represent  the  ratio  between  the 
lateral  and  direct  strains,  and  let  it  be  supposed  to  be  the  same 
in  all  directions.  If  /,  as  in  Art.  2,  represents  the  direct  strain, 
the  edges  of  the  cube  will  become,  by  the  tension  :  a(\  +  /), 
a(i  —rt)  and  a(i  -  rl\  Consequently  the  volume  of  the  re- 
sulting parallelepiped  will  be : 


6  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  4. 

if  powers  of  /  higher  than  the  first  be  omitted.  With  r  be- 
tween o  and  y2,  there  will  be  an  increase  of  volume,  but  not 
otherwise. 

If  the  body  is  subjected  to  compression,  the  edges  of  the 
cube  become :  a(\  —  /,),  a(\  +  rj^  and  a(\  +  rj^ ;  while  the 
volume  of  the  parallelepiped  takes  the  value : 

tf'(i -/,)(!  +  r,/I)2=«3[i+/,(2rI-i)]   ...     (2) 

As  before,  the  higher  powers  of  ^  are  omitted.  If  the  vol- 
ume of  the  cube  is  decreased,  r^  must  be  found  between  o 
and  *A. 


Art.  4.— Relation  between  the  Coefficients  of  Elasticity  for  Shearing  and 
Direct  Stress  in  a  Homogeneous  Body. 

A  body  is  said  to  be  homogeneous  when  its  elasticity,  of  a 
given  kind,  is  the  same  in  all  directions. 

Let  Fig.  i  represent  a  body  subjected  to  tension  parallel  to 
CD.  That  oblique  section  on  which  the  shear  has  the  greatest 
A  E  B  intensity  will  make  an  angle  of 

45°  with  either  of  those  faces 
whose  traces  are  CD  or  BD  ;  for 
i&  OL  is  the  angle  which  any 
oblique  section  makes  with  BD, 
^D  P  the  total  tension  on  BD,  and 
A'  the  area  of  the  latter  surface, 
the  total  shear  on  any  section  whose  area  is  A1  sec  a,  will  be 
P  sin  a.  Hence  the  intensity  of  shear  is  : 


_  .._ _ 


P  sin  a       P  ,  . 

=  —r,  sin  a  cos  a  ......     (i) 


The  second  member  of  Eq.  (i)  evidently  has  its  greatest 


9  = 


Art.  4.]  SHEARING  AND  DIRECT  STRESS.  ^ 

value   for  a  —  45°.     Hence,  if  the  tensile  intensity  on  BD  is 
p 

represented  by  -r  =Xxthe  greatest  intensity  of  shear  will  be  : 
A 


Then  by  Eq.  (3)  of  Art.  2 : 


In  Fig.  I  EK  and  KG  are  perpendicular  to  each  other,  while 
they  make  angles  of  45°  with  either  AB  or  CD.  After  stress, 
the  cube  EKGH  is  distorted  to  the  oblique  parallelepiped 
E'KG'H'.  Consequently  EKGH  and  E'KG'H'  correspond  to 
ABCD  and  AEFD,  respectively,  of  Fig.  I,  Art.  2.  The  angu- 
lar difference  EKG  —  E  KG'  is  then  equal  to  cp  ;  and  EKE' 

=  GKG  =  -£.    Also  EKF  =  45°  -  •?• 

2  2 

Using,  then,  the  notation  of  the  preceding  Arts.,  there  will 
result,  nearly  : 


remembering  that  F'K=  FK(i  +  /)  ;   and  that 
E'F'  =  FK(i  -  rt). 

From   a  trigonometrical   formula,  there  is  obtained,  very 
nearly  : 


tan  45°  +  tan  -|-       I  + 


• 


ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  5. 
From  Eqs.  (4)  and  (5)  : 

V  =  l(i  +  r)      .......     (6) 

Substituting  from  Eq.  (3),  as  well  as  from  Eq.  (i)  of  Art.  2  : 


It  has  already  been  seen  in  the  preceding  Art.  that  r  must 
be  found  between  o  and  j£,  consequently  the  coefficient  of  elas- 
ticity for  shearing  lies  between  the  values  of  %  and  ]/?,  of  that  of 
the  coefficient  of  elasticity  for  tension. 

This  result  is  approximately  verified  by  experiment. 

Since  precisely  the  same  form  of  result  is  obtained  by 
treating  compressive  stress,  instead  of  tensile,  there  will  be 
found,  by  equating  the  two  values  of  G  : 


i  +  r       j  _j_  rj  E       i  +  r 

It  is  clear,  from  the  conditions  assumed  and  operations 
involved,  that  the  relations  shown  by  Eqs.  (7)  and  (8)  can  only 
be  approximate. 


Art.  5. — Expressions  for  Tangential  and  Direct  Stresses  in  Terms  of  the 
Rates  of  Strains  at  any  point  of  a  Homogeneous  Body. 

Let  any  portion  of  material,  perfectly  homogeneous,  be 
subjected  to  any  state  of  stress  whatever.  At  any  point  as  O, 
Fig.  i,  let  there  be  assumed  any  three  rectangular  co-ordinate 
planes ;  then  consider  any  small  rectangular  parallelepiped 
whose  faces  are  parallel  to  those  planes.  Finally  let  the 
stresses  on  the  three  faces  nearest  the  origin  be  resolved  into 


Art.  5.] 


IN   TERMS  OF  STRAINS. 


components   normal    and    parallel  to   their  planes   of  action, 
whose  directions  are  parallel  to  the  co-ordinate  axis. 

The  intensities  of  these  tangential  and  normal  components 
will  be  represented  in  the  usual  manner,  i.e.,  pxy  signifies  a 
tangential  intensity  on 
a  plane  normal  to  the 
axis  of  X  (plane  ZY\ 
whose  direction  is  paral- 
lel to  the  axis  of  F, 
while  pxx  signifies  the 
intensity  of  a  normal 
stress  on  a  plane  nor- 
mal to  the  axis  of  X 
(plane  ZY)  and  in  the 
direction  of  the  axis  of 
X '.  Two  unlike  sub- 
scripts, therefore,  indi- 
cate a  tangential  stress,  while  two  of  the  same  kind  signify  a 
normal  stress. 

From   Eq.  (3)  of  Art.  2  and   Eq.  (7)  of  Art.  4,  there  is  at 
once  deduced  : 

E 

S  =  —, r 

2(1  +  r) 


(0 


Now  when  the  material  is  subjected  to  stress  the  lines 
bounding  the  faces  of  the  parallelepiped  will  no  longer  be  at 
right  angles  to  each  other.  It  has  already  been  shown  in  Art. 
2  that  the  angular  changes  of  the  lines,  from  right  angles,  are 
the  characteristic  shearing  strains,  which,  multiplied  by  G,  give 
the  shearing  intensities. 

Let  £>j  be  the  change  of  angle  of  the  boundary  lines 
parallel  to  ^f  and  Y. 

Let  cp2  be  the  change  of  angle  of  the  boundary  lines 
parallel  to  Y  and  Z. 


IO  ELASTICITY  IN  AMORPHOUS   SOLID  BODIES.     [Art.  5. 

Let    <p3   be   the    change    of   angle    of   the   boundary   lines 
parallel  to  Z  and  X. 

Eq.  (i)  will  then  give  the  following  three  equations : 


(3) 


In  Fig.  i  let  the  rectangle  agfh  represent  the  right  pro- 
jection of  the  indefinitely  small  parallelepiped  dx  dy  dz.  If 
u,  v  and  w  are  the  strains,  parallel  to  the  axis  of  x,  y  and  z,  of 

.   .     ,       .   .   ,     1  ,        .     .         .         .    du   dv  dw 

the  original  point  //,  the  rates  ot  variation  of  strain  —  ,  —  ,  -y-, 

dx  dy    dz 

etc.,  may  be  considered  constant  throughout  this  parallele- 
piped ;  consequently  the  rectangular  faces  will  change  to 
oblique  parallelograms.  The  oblique  parallelogram  dkck,  whose 
diagonals  may  or  may  not  coincide  with  those  of  agfh,  there- 
fore, may  represent  the  strained  condition  of  the  latter  figure. 

Then,  by  Art.  2,  the  difference  between  dhc  and  the  right 
angle  at  h  will  represent  the  strain  cpr  But,  from  Fig.  i,  <px 
has  the  following  value  : 

0>x  =  dhe  +  bhc    .......     (5) 

But  the  limiting  values  of  the  angles  in  the  second  member 
are  coincident  with  their  tangents  ;  hence  : 

de        be  ,-> 


Art.  5.]  STRESSES  IN   TERMS  OF  STRAINS.  II 

But,  again,  de  is  the  distortion  parallel  to  OX  found  by 
moving  parallel  to  OY,  only  ;  hence  it  is  a  partial  differential  of 
u,  or,  it  has  the  value  : 


In  precisely  the  same  manner  be  is  the  partial  differential 
of  v  in  respect  to  #,  or  : 

dv    , 

be  —  -j-  dx. 
dx 

By  the  aid  of  these  considerations,  Eq.  (6)  takes  the  form  : 

du      dv 
V^dy  +  dx     ....... 

If  XY  be  changed  to  XZy  and  then  to  ZX,  there  may  be 
at  once  written  by  the  aid  of  Eq.  (8)  : 


dv  .  dw 
-j-  +  ~T- 
dz  dy 


<z>2  =  -j-      ~T-      .......     (9) 


*=+,  .......  (IO) 

Eqs.  (2),  (3)  and  (4)  now  take  the  following  form  : 


<"> 


(I3) 


12  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  5. 

The  direct  stresses  are  next  to  be  given  in  terms  of  the 
displacements  «,  v  and  w.  Again,  let  the  rectangular  parallele- 
piped dx  dy  dz  be  considered.  Eq.  (i),  of  Art.  i,  shows  that 
the  strain  per  unit  of  length  is  found  by  dividing  the  intensity 
of  stress  by  the  coefficient  of  elasticity,  if  a  single  stress  only 
exists.  But  in  the  present  instance,  any  state  of  stress  what- 
ever is  supposed.  Consequently  the  strain  caused  by/^,  for 
example,  acting  alone  must  be  combined  with  the  lateral 
strains  induced  by/^  and/,,.  Denoting  the  actual  rates  of 
strain  along  the  axes  of  X,  Y  and  Z  by  /z,  /2  and  /3,  therefore, 
the  following  equations  may  be  at  once  written  by  the  aid  of 
the  principles  given  in  Art.  3  : 


= 


, 


Eliminating  between  these  three  equations  : 


.    •    •    09) 


But  if  u,  v  and  w  are  the  actual  strains  at  the  point  where 


'A 

NIVEESIT 

\\ 

Art.  5]          STXESSZS  IN  TERMS  OF 


these  stresses  exist,  the  rates  of  strain  /„  /2  and  /3  will  evi- 
dently be  equal  to^^and^,  respectively.  The  volume 
of  the  parallelepiped  will  be  changed  by  those  strains  to 


dx(i  +  /,XXi  +  4)^(1  +  /3)  =  dx  dy  ds(i  +  /,  +  /,+  /3), 

if  powers  of  /„  /2  and  /3  above  the  first  be  omitted.  The 
quantity  (/x  -f  /a  +  /3)  is,  then,  ///<?  r^  of  variation  of  volume,  or 
/^^  amount  of  variation  of  volume  for  a  cubic  unit.  If  there  be 
put 

0       du        dv        dw 

=       +  ' 


Eqs.  (17),  (18)  and  (19)  will  take  the  forms  : 


~  du 

pxx  =  -        -  6  -f  2G  -=-    .....     (20) 
*  .       i  —  2r  dx 


The  form  in  which  Eqs.  (14),  (i  5)  and  (16)  are  written, 
shows  that  if  pxx,  pyy  or  /„,  is  positive,  the  stress  is  tension, 
and  compression  if  it  is  negative.  Consequently  a  positive 
value  for  any  of  the  intensities  in  Eqs.  (20),  (21)  or  (22)  will  in- 
dicate a  tensile  stress,  while  a  negative  value  will  show  the 
stress  to  be  compressive. 

The  Eqs.  (14)  to  (19),  together  with  the  elimination  in- 
volved, also  show  that  the  coefficients  of  elasticity  for  tension 


14  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  6. 

and  compression  have  been  taken  equal  to  each  other,  and  that 
the  ratio  r  is  the  same  for  tensile  and  compressive  strains. 

Further,  in  Eqs.  (n),  (12)  and  (13),  it  has  been  assumed 
that  G  is  the  same  for  all  planes. 

Hence  Eqs.  (11),  (12),  (13),  (20),  (21)  and  (22)  apply  only  to 
bodies  perfectly  homogeneous  in  all  directions. 

It  is  to  be  observed  that  the  co-ordinate  axes  have  been 
taken  perfectly  arbitrarily. 


Art.  6. — General  Equations  of  Internal  Motion  and  Equilibrium. 

In  establishing  the  general  equations  of  motion  and  equi- 
librium, the  principles  of  dynamics  and  statics  are  to  be  applied 
to  the  forces  which  act  upon  the  parallelepiped  represented  in 
Fig.  i,  the  edges  of  which  are  dx,  dy  and  dz.  The  notation  to 
be  used  for  the  intensities  of  the  stresses  acting  on  the  dif- 
ferent faces  will  be  the  same  as  that  used  in  the  preceding 
Article. 

Let  the  stresses  which  act  on  the  faces  nearest  the  origin 
be  considered  negative,  while  those  which  act  on  the  other 
three  faces  are  taken  as  positive. 

The  stresses  which  act  in  the  direction  of  the  axis  of  X  are 
the  following : 

On  the  face  normal  to  X,  nearest  to       O  ;  —pxx  dy  dz. 

"         "     farthest  from  <9;  (pxx  +  ^j^dx\dy  dz. 

"         "     dy  dx  nearest  to  O ;  —  p^dy  dx. 

"     farthest  from  0 

"         "     dz  dx  nearest  to  O\  —pyxdzdx. 

"     farthest  from  O  ;  (pyx ddz  dx. 


Art.  6.]    EQUA  TIONS  IN  RECTANGULAR  CO-ORDINA  TES. 


(I: 


The  differential  coefficients  of  the  intensities  are  the  rates 
of  variation  of  those  intensities  for  each  unit  of  the  variable, 
which,  multiplied  by  the  dif- 
ferentials of  the  variables, 
give  the  amounts  of  varia- 
tion for  the  different  edges 
of  the  parallelepiped. 

Let  X0  be  the  external 
force  acting  in  the  direction 
of  X  on  a  unit  of  volume  at 
the  point  considered  ;  then 
XQ  dx  dy  dz  will  be  the  ^ 

amount    of    external    force     z    .^'''^ 
acting  on  the  parallelepiped.  ^ 

These  constitute  all  the  forces  acting  on  the  parallele- 
piped inthe  direction  of  the  axis  of  X.,  and  their  sum,  if  un- 
balanced, must  be  equal  to  m  -j-d*dyd*\  in  which  ;;/  is  the 

mass  or  inertia  of  a  unit  of  volume,  and  dt  the  differential  of 
the  time.  Forming  such  an  equation,  therefore,  and  dropping 
the  common  factor  dx  dy  dz,  there  will  result : 


dp x*    ,    dpyx       dpt 
~       '     ~       '      <>» 


~h  ^o  — 


Changing  x  to  j,  y  to  *,  and  z  to  *,  Eq.  (i)  will  become 


dp,,    ,   &3  ,   *s  +  yo  =  m  ™. 
~~~dx         dy        dz  df 


Again,  in  Eq.  (i),  changing  x  to  *,  2  to  y,  and  y  to  x 


1  6  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  6. 


The  line  of  action  of  the  resultant  of  all  the  forces  which 
act  on  the  indefinitely  small  parallelepiped,  at  its  limit,  passes 
through  its  centre  of  gravity,  consequently  it  is  subjected  to  the 
action  of  no  unbalanced  moment.  The  parallelepiped,  therefore, 
can  have  no  rotation  about  an  axis  passing  through  its  centre 
of  gravity,  whether  it  be  in  motion  or  equilibrium.  Hence,  let 
an  axis  passing  through  its  centre  of  gravity  and  parallel  to  the 
axis  of  X,  be  considered.  The  only  stresses,  which,  from  their 
direction  can  possibly  have  moments  about  that  axis,  are  those 
with  the  subscripts  (yz),  (zy),  (yy),  or  (zz).  But  those  with  the 
last  two  subscripts  act  directly  through  the  centre  of  the  paral- 
lelepiped, consequently  their  moments  are  zero.  The  stresses 

~^-dy  .  dx  dz  and  ~~y-dz  .  dx  dy  are  two  of  six  forces  whose 
dy  dz 

resultant  is  directly  opposed  to  the  resultant  of  those  three 
forces  which  represent  the  increase  of  the  intensities  of  the 
normal,  or  direct,  stresses  on  three  of  the  faces  of  the  parallelo- 
piped  ;  these,  therefore,  have  no  moments  about  the  assumed 
axis.  The  only  stresses  remaining  are  those  whose  intensities 
are  /ay  and/^.  The  resultant  moment,  which  must  be  equal 
to  zero,  then,  has  the  following  value: 

pyzdx  dz  .  dy  +  pgydx  dy  .  dz  =  o     ...     .     (4) 

.'•     Py,=   -P,y    .........        (5) 

Hence  the  two  intensities  are  equal  to  each  other. 

The  negative  sign  in  Eq.  (5)  simply  indicates  that  their 
moments  have  opposite  signs  or  directions  ;  consequently,  that 
the  shears  themselves,  on  adjacent  faces,  act  toward  or  from 
the  edge  between  those  faces.  In  Eqs.  (i),  (2)  and  (3),  the 


Art.  6.]   EQ  UA  TIONS  IN  RECTANGULAR  CO-ORDINA  TES.  1 7 

tangential  stresses,  or  shears,  are  all  to  be  affected  by  the  same 
sign,  since  direct,  or  normal,  stresses  only  can  have  different 
signs. 

The    Eq.    (5)   is   perfectly    general,   hence    there    may   be 
written  : 

Pxy  =  pyxy     and    /„  =  pxs (6) 

Adopting  the  notation  of  Lam6,  there  may  be  written  : 
PXX  =  N»        pyy  =  N2,        pu  =  Ny 
P*y  =  T»         px>  =  T2J        pxy  =  Ty 
by  which  Eqs.  (i),  (2)  and  (3)  take  the  following  forms : 


dx          dy          dz 
dT2 


dx         dy         dz     "     °       m  dt*   '     '     '     '     ^J 

-  4-  Y  —  m-—  (K\ 

*°  ~   m  j*2  •    •    •    •    Vs) 

dx     '     dy     '    ~dz      '    ^°  ~  '"^   '     '     '     '     (9) 

The  equations  (11),  (12),  (13),  (20),  (21)  and  (22)  of  the  pre- 
ceding Art.  are  really  kinematical  in  nature  ;  in  order  that  the 
principles  of  dynamics  may  hold,  they  must  satisfy  Eqs.  (7),  (8) 
and  (9).  As  the  latter  stand,  by  themselves,  they  are  applica- 
ble to  rigid  bodies  as  well  as  elastic  ones ;  but  when  the  values 
of  N  and  T,  in  terms  of  the  strains  u,  v  and  w,  have  been  in- 
serted they  are  restricted,  in  their  use,  to  elastic  bodies  only. 
With  those  values  so  inserted,  they  form  the  equations  on 
which  are  based  the  mathematical  theory  of  sound  and  light 
vibrations,  as  well  as  those  of  elastic  rods,  membranes,  etc. 
2 


1  8  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  6. 

In  general,  they  are  the  equations  of  motion  which  the  dif- 
ferent parts  of  the  body  can  have  in  reference  to  each  other, 
in  consequence  of  the  elastic  nature  of  the  material  of  which 
the  body  is  composed. 

If  all  parts  of  the  body  are  in  equilibrium  under  the  action 
of  the  internal  stresses,  the  rates  of  variation  of  the  strains 


,  L      and         ,  will  each  be  equal  to  zero.     Hence,  Eqs.  (;), 
ctt     ctt  dt 

(8)  and  (9)  will  take  the  forms  : 


dT,       dT,       dN, 


These  are  the  general  equations  of  equilibrium.  As  they 
stand,  they  apply  to  a  rigid  body.  For  an  elastic  body,  the 
values  of  N  and  T  from  the  preceding  Art.,  in  terms  of  the 
strains  u,  v  and  w,  must  satisfy  these  equations. 

The  Eqs.  (10),  (n)  and  (12)  express  the  three  conditions  of 
equilibrium  that  the  sums  of  the  forces  acting  on  the  small 
parallelepiped,  taken  in  three  rectangular  co-ordinate  direc- 
tions, must  each  be  equal  to  zero.  The  other  three  conditions, 
indicating  that  the  three  component  moments  about  the  same 
co-ordinate  axes  must  each  be  equal  to  zero,  are  fulfilled  by 
Eqs.  (5)  and  (6).  The  latter  conditions  really  eliminate  three 
of  the  nine  unknown  stresses.  The  remaining  six  conse- 
quently appear  in  both  the  equations  of  motion  and  equilib- 
rium. 

The  equations  (7)  to  (12),  inclusive,  belong  to  the  interior 


Art.  6.]     EQUATIONS  IN  RECTANGULAR  CO-OFDINATES.  19 

of  the  body.  At  the  exterior  surface,  only  a  portion  of  the 
small  parallelepiped  will  exist,  and  that  portion  will  be  a 
tetrahedron,  the  base  of  which  forms  a  part  of  the  exterior 
surface  of  the  body,  and  is  acted  upon  by  external  forces.  Let 
da  be  the  area  of  the  base  of  this  tetrahedron,  and  let/,  q  and 
r  be  the  angles  which  a  normal  to  it  forms  with  the  three  axes 
of  X,  Y  and  Z,  respectively.  Then  will 

da  cos  p  —  dy  dz,  da  cos  q  =  dz  dx,  and  da  cos  r  —  dx  dy. 

Let  P  be  the  known  intensity  of  the  external  force  acting  on 
da,  and  let  TT,  %  and  p  be  the  angles  which  its  direction  makes 
with  the  co-ordinate  axes.  Then  there  will  result : 

X0  =  Pda  .  cos  n,    Y0  =  Pda  .  cos  x  and  Z0  =  P  da  .  cos  p. 

The  originMs  now  supposed  to  be  so  taken  that  the  apex  of 
the  tetrahedron  is  located  between  it  and  the  base ;  hence  that 
part  of  the  parallelepiped  in  which  acted  the  stresses  involving 
the  derivatives,  or  differential  coefficients,  is  wanting ;  con- 
sequently those  stresses  are  also  wanting. 

The  sums  of  the  forces,  then,  which  act  on  the  tetrahedron, 
in  the  co-ordinate  directions,  are  the  following  : 

-  (N,  dy  dz  +  T3  dz  dx  +  T2  dy  dx)  +  Pda  cos  n  =  o, 
-  (T^dz  dy  +  N2  dz  dx  +  Tt  dy  dx)  +  Pda  cos  x  =  o, 

-  (T.dzdy  +  TTdz  dx  +  N^dy  dx)  +  Pda  cos  p  =  o. 

Substituting  from  above  : 

N,  cosp  +  r3  cos  q  +  T2  cos  r  =  P  cos  n      .     .     (13) 


20  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  /. 

Tz  cos  p  -f-  N2  cos  q  -\-  Tl  cos  r  —  P  cos  x      .     .     (14) 

Ta  cos  p  +  T^  cos  q  +  Nz  cos  r  =  P  cos  p      .     .     (i  5) 

These  equations  must  always  be  satisfied  at  the  exterior 
surface  of  the  body;  and  since  the  external  forces  must  always 
be  known,  in  order  that  a  problem  may  be  determinate,  they 
will  serve  to  determine  constants  which  arise  from  the  in- 
tegration of  the  general  equations  of  motion  and  equilibrium. 


Art.  7. — Equations  of  Motion  and   Equilibrium   in   Semi-Polar 
Co-ordinates. 

For  many  purposes  it  is  convenient  to  have  the  conditions 
of  motion  and  equilibrium  expressed  in  either  semi-polar  or 
polar  co-ordinates ;  the  first  form  of  such  expression  will  be 
given  in  this  Article. 

The  general  analytical  method  of  transformation  of  co- 
ordinates may  be  applied  to  the  equations  of  the  preceding 
Article,  but  the  direct  treatment  of  an  indefinitely  small  por- 
tion of  the  material,  limited  by  co-ordinate  surfaces,  possesses 
many  advantages.  In  Fig.  I  are  shown  both  the  small  portion 
of  material  and  the  co-ordinates,  semi-polar  as  well  as  rectangu- 
lar. The  angle  made  by  a  plane  normal  toZY,  and  containing 
OX,  with  the  plane  XY  is  represented  by  <p ;  the  distance  of 
any  point  from  OX,  measured  parallel  to  ZY,  is  called  r\  the 
third  co-ordinate,  normal  to  r  and  9?,  is  the  co-ordinate  x,  as 
before.  It  is  important  to  observe  that  the  co-ordinates  x,  r 
and  <T>,  at  any  point,  are  rectangular. 

The  indefinitely  small  portion  of  material  to  be  considered 
will,  as  shown  in  Fig.  I,  be  limited  by  the  edges  d,r,  dr  and 
r  dcp.  The  faces  dx  dr  are  inclined  to  each  other  at  the  angle 


Art.  7.]     EQUATIONS  IN  SEMI-POLAR   CO-ORDINATES. 


21 


The  intensities  of  the  normal  stresses  in  the  directions  of 
X  and  r  will  be  indicated  by  N^  and  R,  respectively.  The 
remainder  of  the  notation 
will  be  of  the  same  gen- 
eral character  as  that  in 
the  preceding  Article  ; 
z>.,  Txr  will  represent  a 
shear  on  the  face  dr.  r  dcp 
in  the  direction  of  r,  while 
NW  is  a  normal  stress,  in 
the  direction  of  <p,  on  the 
face  dx  dr. 

The  strains  or  dis- 
placements, in  the  direc- 
tions of  x,  r  and  cp,  will 
be  represented  by  u,  p 
and  w  ;  consequently  the 


Fig.1 


unbalanced  forces  in  those  directions,  per  unit  of  mass,  will  be  : 


m 


df< 


-      and  m 


Those  forces  acting  on  the  faces  hf,  fe,  and  he,  will  be  con 
sidered  negative  ;  those  acting  on  the  other  faces,  positive. 


Forces  acting  in  the  direction  of  r. 
R  .  r  dcp  dx,  and  ; 

Rr  dcp  dx 


—  T$rdr  dx,  and  ; 


22  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  7. 

+  T^rdrdx  +  ~*  dcp.drdx. 

—  Txr  .  r  dcp  dr,  and  ; 

+  Txr  .  r  d<p  dr  +  ~^dx  .  r  dcpdr. 
dx 

On  the  face  dr  dx,  nearest  to  ZOX,  there  acts  the  normal 


stress  (N^dr  dx  -f-  '-7  ^9-  dr  <?*)  —  W.  Now  N'  has  a  com- 
ponent acting  parallel  to  the  face/>  and  toward  OX,  equal  to 
N'  sin  (dq>)  =  N'  T—^-  =  N'dcp.  But  the  second  term  of  this 

product  will  hold  (dq>)2,  hence  it  will  disappear,  at  the  limit,  in 
the  first  derivative  of  N'dcp  /.  N'dcp  =  N^dcpdrdx.  Since 
this  force  must  be  taken  as  acting  toward  OX,  it  acts  with  the 
normal  forces  on  ///",  and,  consequently,  must  be  given  the 
negative  sign. 

If  R0  is  the  external  force  acting  on  a  unit  of  volume, 
another  force  (external)  acting  along  r  will  be  R0  .  r  dg)  dr  dx. 

The  sum  of  all  these  forces  will  be  equal  to 

m  .  r  dcp  dr  dx  .  ——. 


Forces  acting  in  the  direction  of  cp. 
dr  dx,  and  ; 


drdx  +  dcp  .  drdx. 

.  r  dcp  dx,  and  ; 


'  Art.  7.]         EQUATIONS  IN   SEMI-POLAR   CO-ORDINATES.  2$ 


dcp  dx. 


—  Tx$  .  r  dcp  dr,  and  ; 

/t7* 

+  TX*  .rdcpdr  +  —f^dx  .  r  dcp  dr. 

As  in  the  case  of  N^,  in  connection  with  the  forces  along 
r,  so  the  force  T$r  dr  dx  has  a  component  along  cp  (normal  to 
fe)  equal  to  T$rdr  dx  .  sin  (dcp)  =  T$rdcpdrdx.  It  will  have 
a  positive  sign,  because  it  acts  from  OX. 

The  external  force  is,  #0  .  r  dcp  dr  dx. 

Forces  acting  in  the  direction  of  x. 

—  NI  .  r  dcp  dry  and  ; 


+  N^r  dcp  dr  +  ~±  dx  .  r  dcp  dr. 
dx 

—  Trx  .  dx  r  dcp,  and 


—  T$xdx  dr,  and; 

+  T^dx  dr  +  d-^-  dcp  .  dx  dr. 

The  external  force  is,  XQ  .  r  dcp  dx  dr. 

Putting  each  of  these  three  sums  equal  to  the  proper  rates 


24  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  7. 

of  variation  of  momentum,  and  dropping  the  common  factor, 
r  dtp  dx  dr  : 


dTr,         dT<,x        Trx  .         d'u 


dTxr         dR            dT^r   ,    R  —  N$$       i     ;?   -      *  **'P     (<z\ 
+  -3T     +  r^r-  H :  +  ^o  -  m  —,—    (3) 


<&  dr 


u,j.rh  u-j-v 0A     .       /r<J>  ~T    *rA      i      ,?;  #22£/      /    \ 

«,    +  -5T-  +  rV  +      ^      +  *•  ~  w-^   (4) 

These  are  the  general  equations  of  motion  (vibration)  in 
terms  of  semi-polar  co-ordinates ;  if  the  second  members  are 
made  equal  to  zero,  they  become  equations  of  equilibrium. 
Eqs.  (2),  (3),  and  (4)  are  not  dependent  upon  the  nature  of  the 
body. 

Since  x,  r,  and  cp  are  rectangular,  it  at  once  follows  that  : 

1 rx  —-  1  xr,  1  r$  =  1 $ry  and  1  xb  --  1  $*•      •     .     .     (5) 

In  order  that  Eqs.  (2),  (3),  and  (4)  may  be  restricted  to 
elastic  bodies,  it  is  necessary  to  express  the  six  intensities  of 
stresses  involved,  in  terms  of  the  rates  of  variation  of  the  strains 
in  the  rectangular  co-ordinate  directions  of  x,  r,  and  (p.  Since 
these  co-ordinates  are  rectangular,  the  Eqs.  (11),  (12),  (13),  (20), 
(21),  and  (22)  of  Article  5,  may  be  made  applicable  to  the  pres- 
ent case  by  some  very  simple  changes  dependent  upon  the 
nature  of  semi-polar  co-ordinates. 

For  the  present  purpose  the  strains  in  the  co-ordinate  direc- 
tions of  x,  yy  and  z  will  be  represented  by  u',  v',  and  w'.  Since 
the  axis  of  x  remains  the  same  in  the  two  systems,  evidently : 

du'        du 

dx  ~~  dx 


Art.  7.]     EQUATIONS  IJV  SEMI-POLAR   CO-ORDINATES.'  2$ 

From  Fig.  I  it  is  clear  that  the  axis  of  y  corresponds  exactly 
to  the  co-ordinate  direction  r ;  hence  : 

dv___dp 
dy       dr 

From  the  same  Fig.  it  is  seen  that  the  axis  of  z  corresponds 
to  <p,  or  rep.  But  the  total  differential,  dw' ,  must  be  considered 
as  made  up  of  two  parts ;  consequently  the  rate  of  variation 

—  will  consist  of  two  parts  also.     If  there  is  no  distortion  in 
dz 

the  direction  of  r,  or  if  the  distance  of  a  molecule  from  the 

...  ,       dw          dw      ,, 
origin  remains  the  same,  one  part  will  be  -^ — .   =  -^— .     If, 

however,  a  unit's  length  of  material  be  removed  from  the  dis- 
tance r  to  r  -\-  p  from  the  centre  (9,  Fig.  I,  while  cp  remains 

constant,  its  length  will  be  changed  from  I  to  I  .  (  i  -f  ~\  in 

which  p  may  be  implicitly  positive  or  negative.     Consequently 
there  will  result : 

dw'         dw         p 
dz        rdg>        r 

For  the  reasons  already  given,  there  follow : 

du!__du  ,      drf_  _  dp 

~dy  ~~  ~dr  dx~~  dx 

In  Fig.  2  let  dc  be  the  side  of  a  distorted  small  portion  of 

the  material,  the  original  position  of  ^ 

which  was  d'e.     Od  is  the  distance  r  ( 
from  the  origin,  ad  =  dr  and  ac  =  dw, 
while  dd'  —  w.     The  angular  change 

in  position  of  dc  is  -    =     — ;  but  an 


26  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  /. 

amount  equal  to  —r=  —  is  due  to  the  movement  of  r,  and  is 
ad       r 

not  a  movement  of  dc  relatively  to  the  material   immediately 
adjacent  to  d. 
Hence : 

dw'        dw        w  dv'          dp 

-j-  —  ~r  ~  — »  also  ~T~  —  ~~r  • 
ay         dr         r  dz         r  acp 

There  only  remain  the  following  two,  which  may  be  at  once 
written : 

dw'      dw          ,     du        du 

—T-—T~      and      ~T~  ~  —j-- 
ax       dx  dz       r  dcp 

The  rate  of  variation  of  volume  takes  the  following  form  in 
terms  of  the  new  co-ordinates  : 


• 

f)  —     -    _j_  ^v'  j_  ^w'  _du    t_dp 


Accenting  the  intensities  which  belong  to  the  rectangular 
system  x,  y,  z,  the  Eqs.  (11),  (12),  (13),  (20),  (21)  and  (22),  of 
Art.  5,  take  the  following  form  : 


i  —  2r 


(7) 


•     (9) 


Art.  8.]  EQUATIONS  IN  POLAR   CO-ORDINATES.  2? 


^P,        ^fdu        dp 

—     23      —    Cr(  — L-     — 

\dr        dx 


dw 


T    = 


If  these  values  are  introduced  in  Eqs.  (2),  (3)  and  (4),  those 
equations  will  be  restricted  in  application  to  bodies  of  homo- 
geneous elasticity  only. 

The  notation  r  is  used  to  indicate  that  the  r  involved  is  the 
ratio  of  lateral  to  direct  strain,  and  that  it  has  no  relation 
whatever  to  the  co-ordinate  r. 

The  limiting  equations  of  condition,  (13),  (14)  and  (15)  of 
Art.  6,  remain  the  same,  except  for  the  changes  of  notation, 
shown  in  Eqs.  (7)  to  (12),  for  the  intensities  N  and  T. 


Art.  8. — Equations  of  Motion  and  Equilibrium  in  Polar  Co-ordinates. 

The  relation,  in  space,  existing  between  the  polar  and 
rectangular  systems  of  co-ordinates  is  shown  in  Fig.  I.  The 
angle  <p  is  measured  in  the  plane  ZY  and  from  that  of  XY\ 
while  tp  is  measured  normal  to  ZY  in  a  plane  which  contains 
OX.  The  analytical  relation  existing  between  the  two  systems 
is,  then,  the  following  : 

x  =  r  sin  rp,    y  —  r  cos  1/1  cos  q>,     and     2  =  r  cos  $  sin  (p. 

The  indefinitely  small  portion  of  material  to  be  considered 
is  a  he  d.  It  is  limited  by  the  co-ordinate  planes  located  by 


28  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  8. 

cp  and  ip,  and  concentric  spherical  surfaces  with  radii  r  and 
r  _j_  dr.  The  directions  r,  cp  and  ^,  at  any  point,  are  rectangu- 
lar ;  hence,  the  sums  of  the  forces  acting  on  the  small  portion 
of  the  material,  taken  in  these  directions,  must  be  found  and 
put  equal  to 


d*ri 


and     ;/z— — , 


in  which  expressions,  p,  rf  and  GO  represent  the  strains  in  the 
direction  of  r,  cp  and  fy  respectively. 


Those  forces  which  act  on  the  faces  ak,  bd  and  cd  will  be 
considered  negative,  and  those  which  act  on  the  other  faces 
positive. 

The  notation  will  remain  the  same  as  in  the  preceding  Ar- 
ticles, except  that  the  three  normal  stresses  will  be  indicated 


Art.  8.]          EQUATIONS  IN  POLAR   CO-ORDINATES. 


Forces  acting  along  r. 
Nr  .  r  dip  r  cosip  dcp. 


dcp-{-  dr  =  r*          dr  +  2r  Nr  dr\ 

cos  ip  dip  d(p. 


-  TV-  r  dip  dr. 


•  r  dip  dr  +  dcp  .  r  dtp  dr. 


—  T^r  .  r  costp  dcp  dr. 

fd(  T^r  cos  ip)   ,  .  ,  dT&r  j  , 

+  TV.  r  cos*p  dcp  dr  +  ^  v    ^        -;  d$  -  cosip  -^  dip 


sin  ip  dtpjr  dcp  dr. 


.  r  dip  dr  .  sin  aOc  =  —  N$  .  r  dip  dr  .  cos  ip  dcp  ; 
on  face  ce. 

.r  cos*p  dcp  dr  .  sin  aOb  =  -  N+  .  r  co  s  i/>  dcp  dr  .  dip  ; 
on  face  be. 


Forces  acting  along  cp. 
—  Tr$  .  r  cosip  dcp  r  dtp. 


cos  $  dip  dcp. 


3O  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  8. 

-  N+  .rdip  dr. 

+  N*  .  r  dip  dr  +  **j[+-d<pr<ty  dr. 

—  TW  .  r  cos  ip  dcp  dr. 


.  rd<p  dr+  d*  =  cos* 


—         sn 


in  ip  dip  jr  dcp  dr. 

« 

r  dfy  dr  .  cos  if>  dcp  ;  on  face  ce. 


-r          j  ,    j    /    •        T.          r  COSib  dcp  \  ~  7,7. 

—  T+j  r  dip  dr  (sin  akc  =  -  /w      )  =  "~  ^V*  *&$&*•  sin  i/>  dcp] 

on  face  ce. 

The  lines  ak  and  ck  are  drawn  normal  to  Oc  and  Oa. 


Forces  acting  along  ty. 
.r  dip. 


dcp  dip  +  ^r  =  r*  ar  +  2r 

cos  ip  dip  dcp. 
dip  dr. 


—  N^  .  r  costy  dcp  dr. 


Art.  8.]  EQUATIONS  IN  POLAR   CO-ORDINATES.  31 


r  cos     d    dr 


—  N^  sin  $  difArdcp  dr. 
4-  T^r  .  r  costy  dcp  dr  .  dip  ;  on  face  be^ 
-f  N$  .  r  d^  dr  .  sin  akc  —  -f-  N$  r  dfy  dr  .  sin  $  dtp  ;  on  face  ce. 

The  volume  of  the  indefinitely  small  portion  of  the  ma- 
terial is  (omitting  second  powers  of  indefinitely  small  quan- 
tities) : 

r  cos  fy  dq>  .  r  dfy  .  dr  =  A  V; 

and  its  mass  is  m  multiplied  by  this  small  volume.  The  latter 
may  be  made  a  common  factor  in  each  of  the  three  sums  to  be 
taken. 

The  external  forces  acting  in  the  directions  R,  cp  and  ^  will 
be  represented  by  : 

RQAV,    $><>AV    and 


respectively. 

Taking  each  of  the  three  sums,  already  mentioned,  and 
dropping  the  common  factor  A  V,  there  will  result  : 


r_  ^r         ,  r 


dr  r  cos  ip  .  dcp        r  dfy  r 

+  £.=  >*         .......    (i) 


dr  r  cos  if>  .  dcp        r 


[  |    ^_^ 


32  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  8. 


dN± 
dr  r  cos  ty  d<p       r  dtp 


Since  r,  cp  and  ^  are  rectangular  at  any  point  : 

T$r  —  Trfr     T^  =  T^r    and     T^  =  T^. 
Hence  : 


+  TV  —  tan  $(T^  +  7^,)  _    3^—2  tan  $  . 
r  r 


+  T^r  -  tan  ^(N*  -  N^)  __  3^-  tan 


These  relations  somewhat  simplify  the  first  members  of 
Eqs.  (2)  and  (3). 

Eqs.  (i),  (2)  and  (3)  are  entirely  independent  of  the  nature 
of  the  material  ;  also,  they  apply  to  the  case  of  equilibrium,  if 
the  second  members  are  made  equal  to  zero. 

The  rectangular  rates  of  strain,  at  any  point,  in  terms  of 
r,  cp  and  ^  are  next  to  be  found.  As  in  the  preceding  Art., 
the  rates  of  strain  in  the  rectangular  directions  of  r,  cp  and  ip 
will  be  indicated  by  : 

<M_    du/_    du'    ^/     M 
df    dz"    ~d3'    dx»   dy"    ( 

Remembering  the  reasoning  in  connection  with  the  value  of 

dw' 

-£-,  in  the  preceding  Art.,  and  attentively  considering  Fig.  I, 

there  may  at  once  be  written  : 


Art.  8.]          EQUATIONS  IN  POLAR   CO-ORDINATES. 


33 


du' 


In  Fig.  i,  \{  ac—\  and  ab  =  GO,  while  ak  =  rcot.ip  (ak  is 
perpendicular  to  aO\  the  difference  in  length  between  ac  and 
bh  will  be  : 


r  cot  i/> 

This  expression  is  negative  because  a  decrease  in  length  takes 
place  in  consequence  of  a  movement  in  the  positive  direction 
of  rip. 

Again,  a  consideration  of  Fig.  i,  and  the  reasoning  con- 
nected with  the  equation  above,  will  give : 

dw'  _          drf  p        oj  tan  $ 

dz'         r  cos  ip  dcp       r  r 

Without  explanation  there  may  at  once  be  written : 
dv'        dp 


Fig.  i  of  this,  and  Fig.  2  of  the  preceding  Art.  give  : 
du         doo        03  dv'          dp 

—    _    _    ctnH  —          ' 

dy'  "  dr         r'  dx'     ~  r  dip' 

These  are  to  be  used  in  the  expression  for  T^     Precisely 
the  same  Figs,  and  method  give  : 

^  -  dp  nd     — '  =  —  —  -• 

~dz'  ~  ~  r  cos  ip  dcp '  dy'        dr       r' 

which  are  to  be  used  in  finding  T^. 
3 


34  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.     [Art.  8. 


The  expression  for  -j-j  will  be  composed  of  the  sum  of  two 

parts.  In  Fig.  2,  ab  is  the  original  position  of  r  dty,  and  after 
the  strain  rj  exists  it  takes  the  position  ec.  Consequently  ac 
(equal  and  parallel  to  bd  and  perpendicular 
to  ak)  represents  the  strain  77,  while  ed  rep- 
resents drf.  Since,  also,  fc  is  perpendicu- 
lar to  ck9  the  strains  of  the  kind  77  change 
the  right  angle  fck  to  the  angle  fee ;  or 
the  angle  eck  is  equal  to 

dw'  ,  ed       ca 

— — -  =  ted  +  dck  =  —  4-  — 
dx  dc       ak 

drf  77 


In  Fig.  2,  the  points  a,  b  and  £  are  iden- 
tical with  the  points  similarly  lettered  in  Fig.  i.  The  expres- 
sion for  ~  may  be  at  once  written  from  Fig.  i.  There  may, 
then,  finally  be  written  : 


dw' 
dx 


drf         rj  tan 

T~         : 


rdtp 


,       du  doo 

and,      -=-,  =  — j-  . 

dz         r  cos  ip  dcp 


These  equations  will  give  the  expression  for 
The  value  of 


dx 


now  takes  the  following  form  : 


dr       r  cos  fy  dcp        r  dip         r 


GO  tan 


35 


Art  8.]          EQUATIONS  IN  POLAR   CO-ORDINATES. 

The  last  two  terms  are  characteristic  of  the  spherical  co- 
ordinates. 

The  equations  (20),  (21),  (22),  (u),  (12)  and  (13),  of  Art.  (5) 
take  the  forms  : 


r  dp 

=   G 


r  cos 


If  these  values  are  inserted  in  Eqs.  (i),  (2)  and  (3),  the 
resulting  equations  will  be  applicable  to  isotropic  material 
only. 

As  in  the  preceding  Art.,  r  is  used  to  express  the  ratio 
between  direct  and  lateral  strains,  and  has  no  relation  what- 
ever to  the  co-ordinate  r. 

It  is  interesting  and  important  to  observe  that  the  equa- 
tions of  motion  and  equilibrium  for  elastic  bodies,  are  only 
special  cases  of  equations  which  are  entirely  independent  of 
the  nature  of  the  material,  of  equations,  in  fact,  which  express 
the  most  general  conditions  of  motion  or  equilibrium. 


CHAPTER    II. 


THICK,  HOLLOW  CYLINDERS  AND  SPHERES,  AND  TORSION. 

Art.  9.— Thick,  Hollow  Cylinders. 

IN  Fig.  I  is  represented  a  section,  taken  normal  to  its  axis, 
of  a  circular  cylinder  whose  walls  are  of  the  appreciable  thick- 
ness /.  Let/  and/j  represent  the  interior  and  exterior  inten- 
sities of  pressures,  respectively.  The  material  will  not  be 
stressed  with  uniform  intensity  throughout  the  thickness  /.  Yet 

if  that  thickness,  comparatively  speak- 
ing, is  small,  the  variation  will  also  be 
small ;  or,  in  other  words,  the  intensity 
of  stress  throughout  the  thickness  / 
may  be  considered  constant.  This 
approximate  case  will  first  be  con- 
sidered. 

The  interior  intensity  /  will  be 
considered  greater  than  the  exterior 
/„  consequently  the  tendency  will 
be  toward  rupture  along  a  diametral 

plane.  If,  at  the  same  time,  the  ends  of  the  cylinder  are  taken 
as  closed,  as  will  be  done,  a  tendency  to  rupture  through  the 
section  shown  in  the  Fig.  will  exist. 

The  force  tending  to  produce  rupture  of  the  latter  kind 
will  be  : 


(I) 


Art  9.]  THICK,  HOLLO  IV  CYLINDERS.  37 

If  TV,  represents  the   intensity  of  stress  developed  by  this 
force, 


Nt  =  :  _  -          - 

n(r?  -  r'*)  r?  -  r'* 

If  the  exterior  pressure  is  zero,  and  if  r  is  nearly  equal  to 


N  -  -     r 

'~  2(rx-r')~'  V 

In  this  same  approximate  case,  the  tendency  to  split  the 
cylinder  along  a  diametral  plane,  for  unit  of  length,  will  be  : 


If  N!  is  the  intensity  of  stress  developed  by  F'  : 


N'  is  thus  seen  to  be  twice  as  great  as  N^  when/,  =  o.  If, 
therefore,  the  material  has  the  same  ultimate  resistance  in  both 
directions  the  cylinder  will  fail  longitudinally  when  the  interior 
intensity  is  only  half  great  enough  to  produce  transverse  rup- 
ture ;  the  thickness  being  assumed  to  be  very  small  and  the  ex- 
terior pressure  zero. 

NI  and  N'  are  tensile  stresses,  because  the  interior  pressure 
was  assumed  to  be  large  compared  with  the  exterior.  If  the 
opposite  assumption  were  made,  they  would  be  found  to  be 
compression,  while  the  general  forms  would  remain  exactly  <the 
same. 


38  THICK,  HOLLOW  CYLINDERS.  [Art.  9. 

The  preceding  formulas  are  too  loosely  approximate  for 
many  cases.  The  exact  treatment  requires  the  use  of  the 
general  equations  of  equilibrium,  and  the  forms  which  they  take 
in  Art.  7  are  particularly  convenient.  As  in  that  Art.,  the  axis 
of  x  will  be  taken  as  the  axis  of  the  cylinder. 

Since  all  external  pressure  is  uniform  in  intensity  and  nor- 
mal in  direction,  no  shearing  stresses  will  exist  in  the  material 
of  the  cylinder.  This  condition  is  expressed  in  the  notation 
of  Art.  7  by  putting  : 


Again  the  cylinder  will  be  considered  closed  at  the  ends, 
and  the  force  F,  Eq.  (i),  will  be  assumed  to  develop  a  stress 
of  uniform  intensity  throughout  the  transverse  section  shown 
in  Fig.  i.  This  condition,  in  fact,  is  involved  in  that  of  making 
all  the  tangential  stresses  equal  to  zero. 

Since  this  case  is  that  of  equilibrium,  the  equations  (2),  (3) 
and  (4)  of  Art.  7  take  the  following  form,  after  neglecting  XQ, 
R0  and  <2>0  : 

dN, 


dr 


-H  ...... 


These  equations  are  next  to  be  expressed  in  terms  of  the 
strains  u,  p  and  w. 

^In  consequence  of  the  manner  of  application  of  the  external 
forces,  all  movements   of   indefinitely  small   portions   of   the 


Art.  9.]  THICK,  HOLLOW  CYLINDERS.  39 

material  will   be   along  the   radii  and  axis   of  the   cylinder. 
Hence: 

u  will  be  independent  of  r  and  cp; 
P     "     "  "  "  cp    "     x  ; 

w  —  o. 

The  rate  of  change,  therefore,  of  volume  will  be  (Eq.  (6)  of 

Art.  7): 


dx       dr 


(8) 


As  p  is  independent  of  x,  —  =  -j—  ;  hence  if  the  value  of 

NI  be  taken  from  Eq.  (7)  of  Art.  7  and  put  in  Eq.  (5)  of  this 
Art: 


d*u         rd*u  _ 

dx  Kd*  H     **&  = 


.*.     -3—  =  o,     and     u  —  ax  -\-  a  . 

But  the  transverse  section  in  which  the  origin  is  located 
may  be  considered  fixed.  Consequently  if  x  —  o,  u  =  o  and 
thus  a  =  o.  The  expression  for  u  is  then  :  u  =  ax. 

The  ratio  u  ~-  x  is  the  /'  of  Eq.  (i),  Art.  I  ;  while  the  / 
of  the  same  equation  is  simply  Nt  of  Eq.  (2),  given  above. 
Hence  : 


_ 

~  x  ~~  E 


Again,  Eq.  (8),  of  Art.  7,  in  connection  with  Eqs.  (8)  and  (6) 
of  this,  gives  : 


4O  THICK,  HOLLOW  CYLINDERS.  [Art.  9. 


+  ,  2G         +  dP        P_ 

•  "  TJr       r2)          ^U^     "  rdr       r*     " 


^  ^    /  j  i  /         /^          i 

..      *     i    £  =  c.  c 
r  dp  -\-  p  dr  — 


^rr2  cr         b 

T  +    '  or>  p  =  T  +  7"    •    ' 


This  value  of  p  in  Eqs.  (8)  and  (9)  of  Art.  7  will  give 


i  —  2r 


.1  +  .    .    .    (12) 

2  r 


i  —  2r         2 


At  the  interior  surface  R  must  be  equal  to  the  internal 
pressure,  and  at  the  exterior  surface  to  the  external  pressure. 
Or,  since  negative  signs  indicate  compression  ; 

If  r  =  r' R-  -p. 

If  r  =  r, R  =  -  A. 

Either  of  these  equations  is  the  simple  result  of  applying 
Eqs.  (13),  (14)  and  (15)  to  the  present  case,  for  which, 


Art.  9.]  THICK,  HOLLOW  CYLINDERS.  41 

cosp  —  cos  r  =  cos  n  =  cos  p  =  o, 
cos  q  —  cos  x  =  i,     and     P  =  —  p     or     -  /,. 
Applying  Eq.  (i  i)  to  the  two  surfaces : 

-p=2G^±^+L-^    .    .    .    (I3) 


Subtracting  (14)  from  (13): 


Inserting  this  value  in  Eq.  (13): 


The  general  expressions  of  7?  and  N^,  freed  from  the  arbi- 
trary constants  of  integration,  can  now  be  easily  written  by 
inserting  these  last  two  values  in  Eqs.  (i  i)  and  (12).  By  making 
the  insertions  there  will  result  : 

_    PS*  _  p*  _  (A  -  p)  r*r»        i_ 
">'•  -  r?  r"  -r?  r*  ' 

#„  -  IS*-**  +  (A-/)^2  .    1  ,  (16) 

r'«  _  r?  r*  -  rt*  r3 

The  stress  N     is  a  tension  directed  around  the  cylinder,  and 


42  THICK,  HOLLOW  CYLINDERS.  [Art.  9. 

has  been  called  "  hoop  tension."  Eq.  (16)  shows  that  the  hoop 
tension  will  be  greatest  at  the  interior  of  the  cylinder.  An  ex- 
pression for  the  thickness,  /,  of  the  annulus  in  terms  of  the 
greatest  hoop  tension  (which  will  be  called  Ji)  can  easily  be 
obtained  from  Eq.  (16). 

If  r  —  r'  in  that  equation  : 

2/y,'  -  p  (r'*  +  rf) 
r'*  -  r? 


2pi- 


Eq.  (17)  will  enable  the  thickness  to  be  so  determined  that 
the  hoop  tension  shall  not  exceed  any  assigned  limit  h.  If/! 
is  so  small  in  comparison  with  /  that  it  may  be  neglected,  / 
will  become  : 


If  /,  is  greater  than  /,  N$$  becomes  compression,  but  the 
equations  are  in  no  manner  changed. 

The  values  of  the  constants  b  and  c  may  easily  be  found 
from  the  two  equations  immediately  preceding  Eq.  (15). 

It  is  interesting  to  notice  that  the  rate  of  change  of  volume, 
By  is  equal  to  (a  +  c)  and,  therefore,  constant  for  all  points. 


Art.  10.] 


TORSION  IN  EQUILIBRIUM. 


43 


Art.   10. — Torsion  in   Equilibrium. 

The  formulas  to  be  deduced  in  this  Article  are  those  first 
given  by  Saint- Venaut,  but,  with  one  or  two  exceptions,  es- 
tablished in  a  different  manner. 

It  will  in  all  cases,  except  that  of  the  final  result  for  a  rec- 
tangular cross  section,  be  convenient  to  use  those  equations  of 
Art.  7  which  are  given  in  terms  of  semi-polar  co-ordinates. 

Let  Fig.  i  represent  a  cylindrical  piece  of  material,  with 
any  cross  section,  fixed  in  the  plane  ZY,  and  let  the  origin  of 
co-ordinates  be  taken  at  O.  Let 
it  be  twisted,  also,  by  a  couple 


\ 


o— s 


P .  ab  =  PI, 

the  plane  of  which  is  parallel  to 
ZY.  The  material  will  thus  be 
subjected,  to  no  bending,  but  to 
pure  torsion. 

The  axis  of  the  piece  is  sup- 
posed to  be  parallel  to  the  axis  of 
Jf  as  well  as  the  axis  of  the  couple. 
Normal  sections  of  the  piece,  orig-  z 
inally  parallel  to  ZOY,  will  not  re-  7  Fig.1 
main  plane  after  torsion  takes  place.  But  the  tendency  to 
twist  any  elementary  portion  of  the  piece  about  an  axis  pass- 
ing through  its  centre  and  parallel  to  the  axis  of  X  will  be  very 
small  compared  with  the  tendency  to  twist  it  about  either  the 
axis  of  r  or  cp  ;  consequently  the  first  will  be  neglected.  In 
the  notation  of  Art.  7,  this  condition  is  equivalent  to  making 
T^  =  o. 

As  the  piece  is  acted  upon   by  a  couple  only,  all  normal 
stresses  will  be  zero. 


44  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

Eqs.  (7),  (8),  (9)  and  (11),  of  Art.  7,  then  become  : 

N   = 


1—21 


R        =  _9  = 

i  —  2r  dr 


G^_        p_     = 
'      - 


—  2r  r 


After  introducing  the  values  of  Trx  and  7^,  from  Eqs.  (10) 
and  (12)  of  Art.  7,  in  Eqs.  (2),  (3)  and  (4)  of  the  same  Article, 
at  the  same  time  making  the  external  forces  and  second  mem- 
bers of  those  equations  equal  to  zero,  and  bearing  in  mind  the 
conditions  given  above,  there  will  result : 


d  1  rx          d  I  $x          1  rx  ,-*f  a  u          a2p  d2w        .      d2u 

dr          r  d<p  r  \  dr2        dr  dx       r  dq>  dx 

du          dp  \ 
+  rJr+^)=° 


dTrx 
dx 


dx  \dx2         r  dtp  dxj 


:    =o.     .     -     .     (7) 


Also  by  Eq.  (6)  of  Art.  7 : 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  45 

Q^dll^         dp         _dw_          p 

dx^    dr  +  r  d     +  r 


The  cylindrical  piece  of  material  is  supposed  to  be  of  such 
length,  that  the  portion  to  which  these  equations  apply  is  not 
affected  by  the  manner  of  application  of  the  couple.  This 
portion  is,  therefore,  twisted  uniformly  from  end  to  end  ;  con- 
sequently the  strain  u  will  not  vary  with  any  change  in  x. 
Hence  : 

du 


Eq.  (i)  then  shows  that  6  =  o.  This  was  to  be  anticipated, 
since  a  pure  shear  cannot  change  the  volume  or  density.  Be- 
cause 8  =  o,  Eqs.  (2)  and  (3)  at  once  give  : 


.        - 

dr        rdcp 


As   the  torsion    is   uniform  throughout   the   portion   con- 
sidered : 


4?  =0=4- 

dx  r  dx 


Eq.  (11)  in  connection  with  Eq.  (10),  gives: 


d'w 


r  dx  dtp 


=  0 (12) 


Eqs.  (11)  and  (12),  in  connection  with  Eq.  (10),  reduce  Eq. 
(5)  to  the  following  form  : 


46  TORSION  IN  EQUILIBRIUM.  [Art.  TO. 

d(   —  \ 
d^        d-u          du     _          d*u  \rdr)  ,    } 

""  dr*         r  dr  ~  dt    "*  dr 


Both  terms  of  the  second  member  of  Eq.  (6)  reduce  to  zero 
by  Eqs.  (9)  and  (n),  and  give  no  new  condition.  The  second 
term  of  the  second  member  of  Eq.  (7)  is  zero  by  Eq.  (9)  ;  the 
remaining  term  therefore  gives  : 


As  the  stress  is  all  shearing,  p  will  not  vary  with  (p. 
Hence  : 


Eqs.  (10),  (n)  and  (15)   show  that  p  —  o,  and  reduce  Eq. 
(4)  to  : 

.  $?  -  ^  =  o  .  (16) 

dr         r 


Eq.  (10)  now  becomes  —  —  =  o,  and  shows  that  w  does  not 


contain  op  ;  while  Eq.  (14)  shows  that  w  does  not  contain  ;r2  or 
any   higher  power  of  x.     The  strain  w,  in    connection  with 
these  conditions>  is  to  be  so  determined  as  to  satisfy  Eq.  (16). 
If  a  is  a  constant,  the  following  form  fulfills  all  conditions  : 

w  =  arx    ........     (17) 

Eq.  (17)  shows  that  the  strain  w,  in  the  direction  of  op,  i.e., 
the  angular  strain  at  any  point,  varies  directly  as  the  distance 


Art.    10.]  TORSION  IN  EQUILIBRIUM.  47 

from  the  axis  of  X,  and,  as  the  distance  from  the  origin  measured 
along  that  axis.  This  is  a  direct  consequence  of  making  T^ 
=  o. 

The  quantity  a  is  evidently  the  angle  of  torsion,  or  the 
angle  through  which  one  end  of  a  unit  of  fibre,  situated  at 
unit's  distance  from  the  axis,  is  twisted  ;  for  if ; 

r  =  x  =  i,     w  —  a. 


An  equation  of  condition  relative  to  the  exterior  surface  of 
the  twisted  piece  yet  remains  to  be  determined  ;  and  that  is 
to  be  based  on  the  supposition  that  no  external  force  whatever 
acts  on  the  outer  surface  of  the  piece.  In  Eqs.  (13),  (14)  and 
(15)  of  Art.  6,  consequently,  P  =  o.  The  conditions  of  the 
problem  also  make  all  the  stresses  except : 

T3  =  Txr     and     T,  =  T^ 

equal  to  zero,  while  the    cylindrical  character   of  the   piece 
makes : 

p  —  90°     /.     cos  p  =  o. 

If  cos  t  be  written  for  cos  r  : 

cos  t  —  sin  q. 

Eq.  (13),  just  cited,  then  gives  : 

Txrcosq+  T^sinq  =  0 (18) 

But  since  p  —  o  and  w  —  arx\ 

TX=G^  •     •     09) 


48  TORSION  IN  EQUILIBRIUM.    '  [Art.  IO. 


and 


Eq.  (18)  now  becomes: 

du 
dr 


=  —  tan  q  = -y—  ....     (21) 

du  r0  a  cp 

—-. \-  ar 

r  dq> 

in  which  r0  is  the  value  of  r  for  the  perimeter  of  any  normal 
section. 

Eqs.  (13)  and  (21)  are  all  that  are  necessary  and  all  that 
exist,  for  the  determination  of  the  strain  u.  Eq.  (13)  must  be 
fulfilled  at  all  points  in  the  interior  of  the  twisted  piece,  while 
Eq.  (21)  must,  at  the  same  time,  hold  true  at  all  points  of  the 
exterior  surface. 

.After  u  is  determined,  Txr  and  Tx$  at  once  result  from  Eqs. 
(19)  and  (20).     The  resisting  moment  of  torsion  then  becomes: 


.  dr  =  G  .rdrd<f>+GaIp  .     (22) 


In  this  equation  Ip  =  llr3  .  r  d<p  dr  is  the  polar  moment  of 

inertia  of  the  normal  section  of  the  piece  about  the  axis  of 
X,  and  the  double  integral  is  to  be  extended  over  the  whole 
section. 

According  to  the  old,  or  common,  theory  of  torsion  : 

M  =  Gal? 

The  third  member  of  Eq.  (22),  shows,  however,  that  such  an 
expression  is  not  correct  unless  u  is  equal  to  zero,  z>.,  unless 
all  normal  sections  remain  plane  while  the  piece  is  subjected 


Art.   10.]  TORSION  IN  EQUILIBRIUM.  49 

to  torsion.     It  will  be  seen  that  this  is  true  for  a  circular  sec- 
tion only. 

It   may  sometimes  be   convenient  to  put   Eq.  (22)  in  the 
following  form  : 

M  =  G\^r  dr  .  ~  d<p  -j-  Galp  =  G\u  .  r  dr  +  G«It .     (23) 


In  this  equation  u  is  to  be  considered  as : 

~j — '  d<j> ; 
Jn  am 


while  the  remaining  integration  in  r  is  to  be  so  made  that  the 
whole  section  shall  be  covered. 

It  is  very  important  to  observe  that  the  equations  of  con- 
dition for  the  determination  of  u,  and  consequently  the  general 
values  of  Txr  and  Tx$,  are  wholly  ^independent  of  any  consider- 
ations regarding  the  position  of  the  axis  of  torsion,  or  the  axis 
of  X.  It  follows  from  this,  that  the  resistance  of  pure  torsion  is 
precisely  the  same  wherever  may  be  the  axis  about  which  the 
piece  is  twisted.  It  is  to  be  borne  in  mind,  however,  that,  if 
the  axis  of  the  twisting  is  not  the  axis  of  the  cylindrical  piece, 
the  latter  will  be  subjected  to  combined  bending  and  torsion  ; 
the  bending  being  produced  by  a  force  sufficient  to  cause  the 
piece  to  take  the  helical  position  ne- 
cessitated by  the  torsion.  The  cylin- 
drical axis  is  the  straight  line  locus  of 
the  centres  of  gravity  of  all  the  normal 
sections. 

If,  as  in  Fig.  2,  there  are  n  cylinders 
whose  centres  c  are  all  at  the  same 
distance  Cc  =  I  from  the  centre  C  of 
twisting,  or  motion ;  and  if  M  is  the 
total  moment  of  torsion  of  the  system, 
4 


50  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

while  m  is  the  moment  of  torsion  of  each  cylinder  about  its 
own  axis  or  centre  <:,  then  will  M  =  nm  ;  and  each  cylinder 
wilt  be  subject  to  a  bending  moment  whose  amount  can  -  be 
determined  from  the  condition  that  the  diameter  of  each  piece 
lying  along  Cc  before  torsion,  must  pass  through  C  after,  and 
during,  torsion,  also. 

Since  Txr  and  Tx$  act  at  right  angles  to  each  other,  the  re- 
sultant intensity  of  shear  at  any  point  in  an  originally  normal 
section  of  the  twisted  piece  will  be : 


T=  VTxr*+  2V    ......     (24) 

According  to  the  ordinary  methods  of  the  calculus,  the  co- 
ordinates of  the  point  at  which  T  has  its  greatest  value  must 
satisfy  the  equations  : 

dT       dT 


_  d*T 

dcp*  \°'     dr* 

After  the  solution  of  Eqs.  (25),  it  will  usually  be  necessary 
only  to  inspect  the  resulting  value  of  Ty  in  order  to  determine 
whether  it  is  a  maximum  or  minimum,  without  applying  the 
tests  which  follow  those  equations. 


Equations  of  Condition  in  Rectangular  Co-ordinates. 

In  the  case  of  a  rectangular  normal  section,  the  analysis  is 
somewhat  simplified  by  taking  some  of  the  quantities  used  in 
terms  of  rectangular  co-ordinates. 

In  the  notation  of  Art.  6,  all  stresses  will  be  zero  except 


V"        OK    THE 

/UNIVERSITY 

Art.   10.]  TORSION  IN 


T3  and  Tv     Hence  Eqs.  (10),  (i  i)  and  (12)  of  that  Article  re- 
duce  to : 

dT3       dT, 


dT, 

-dx  =  °- 

The  strains  in  the  directions  of  x,  y  and  z  are,  respectively, 
u,  v  and  w.  Introducing  the  values  of  T3  and  Ta  in  the  equa- 
tions above,  in  terms  of  these  strains,  from  Eqs.  (11)  and  (13) 
of  Art.  5 ;  and  then  doing  the  same  in  reference  to  the  con- 
ditions 

the  following  equations  will  result : 
d*u        d*u 


dv    .    dw 


The  operations  by  which  these  results  are  reached  are  iden- 
tical with  those  used  above  in  connection  with  semi-polar  co- 
ordinates, and  need  not  be  repeated. 

Eq.  (27)  is  satisfied  by  taking  : 


v  —       ax  z  ; 

w  —  —  axy  ; 


in  which  a  is  the  angle  of  torsion,  as  before. 


52  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

Eqs.  (n)  and  (13)  of  Art.  5  then  give  : 

.     .     .     (28)' 


dx  J 

f    N 

•  •  •  (29) 


The  element  of  a  normal  section  is  dz  dy.     Hence  the  mo- 
ment of  torsion  is 


.-.    M  =  G  [(*«  da  -  yu  dy)  +  Gaff    ....     (31) 


is  the  polar  moment  of  inertia  of  any  section  about  the  axis 
oiX. 

The  integrals  are  to  be  extended  over  the  whole  section  ; 
hence,  in  Eq.  (31),  zu  dz  is  to  be  taken  as  : 


z  az 


dy  as : 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  53 

in  which  expressions,  y0  and  z0  are  general  co-ordinates  of  the 
perimeter  of  the  normal  section. 

Eq.  (26)  is  identical  with  Eq.  (13),  and  can  be  derived  from 
it,  through  a  change  in  the  independent  variables,  by  the  aid 
of  the  relations: 

z  —  r  cos  cp ;  and  y  =  r  sin  (p. 


Solutions  of  Eqs.  (13)  and(2\\ 

It  has  been  shown  that  the  function  u,  which  represents  the 
strain  parallel  to  the  axis  of  the  piece,  must  satisfy  Eq.  (13)  [or 
Eq.  (26)]  for  all  points  of  any  normal  section,  and  Eq.  (21)  (or 
a  corresponding  one  in  rectangular  co-ordinates)  at  all  points 
of  the  perimeter  ;•  and  those  two  are  the  only  conditions  to  be 
satisfied. 

It  is  shown  by  the  ordinary  operations  of  the  calculus  that 
an  indefinite  number  of  functions  u,  of  r  and  q>,  will  satisfy  Eq. 
(13);  and,  of  these,  that  some  are  algebraic  and  some  tran- 
scendental. 

It  is  further  shown  that  the  various  functions  u  which 
satisfy  both  Eqs.  (13)  and  (21)  differ  only  by  constants. 

If  u  is  first  supposed  to  be  algebraic  in  character,  and  if  rn 
c»  c3  etc.,  represent  constant  coefficients,  the  following  general 
function  will  satisfy  Eq.  (13)  : 


( 
*  \ 


Cj  r  sin  cp  -\-  c2r*  sin  2cp  +  c3  r*  sin  3<p  + 
-f  c\  r  cos  cp  -f  c'2r*  cos  2cp  +  c'3  r*  cos  $cp  + 


and  the  following  equation,  which  is  supposed  to  belong  to  the 
perimeter  of  a  normal  section  only,  will  be  found  to  satisfy 


T* 

-  +  c,  r  cos  cp  +  c2  r2  cos  2<p  +  c3  r*  cos  39?  +  .  . 
'  —  c\  r  sin  cp  —  c'2  r3  sin  2(p  —  c'3  r3  sin  $cp  —  .  .  .  =  C  (33) 


54  TORSION  1%  EQUILIBRIUM.  [Art.  IO. 

C  is  a  constant  which  changes  only  with  the  form  of  sec- 
tion. 

If  -r  and  — T-  be  found  from  Eq.  (32),  while ~  be  taken 

dr          rd<p  r0  dq> 

from  Eq.  (33),  and  if  these  quantities  be  then  introduced  in  Eq. 
(21),  it  will  be  found  that  that  equation  is  satisfied. 

The  only  form  of  transcendental  function  needed,  among 
those  to  which  the  integration  of  Eq.  (13)  or  Eq.  (26)  leads, 
will  be  given  in  connection  with  the  consideration  of  pieces 
with  rectangular  section,  where  it  will  be  used. 


Elliptical  Section  about  its  Centre. 

Let  a  cylindrical  piece  of  material  with  elliptical  normal 
section  be  taken,  and  let  a  be  the  semi-major  and  b  the  semi- 
minor  axis,  while  the  angle  cp  is  measured  from  a  with  the 
centre  of  the  ellipse  as  the  origin  of  co-ordinates,  since  the 
cylinder  will  be  twisted  about  its  own  axis.  The  polar  equa- 
tion of  the  elliptical  perimeter  may  take  the  following  shape  : 

r*    ,    r*      b2  —  a*  a*b* 

2+2    •*-+*""  2<P  =  *^r»     •     •    ;    (34> 

By  a  comparison  of  Eqs.  (33)  and  (34),  it. is  seen  that : 


and  that  all  the   other  constants  are  zero.     Hence   Eq.  (32) 
gives : 

b2  -  a*  a 

M  =  a2(a2  +  b2)      Sm  29  =  2  f    Sm  29    '     ' 

The  quantity  represented  by /is  evident. 


Art,  10.]  TORSION  IN  EQUILIBRIUM.  55 


By  Eqs.  (19)  and  (20): 

~          ~  b*  —  a2 


rsm2<p  ......     (36) 


^ 

.     .     .     (37) 


Since  r°  '  r°    9  =  dAy  A  being  the  area  of  the  ellipse,  or 

itab,  the  second  member  of  Eq.  (22),  by  the  aid  of  Eq.  (37), 
may  take  the  form  : 


M  =  Gad(P—-     r*  cos  2(P 


Then  using  Eq.  (34)  : 


If  Ip  is  the  polar  moment  of  inertia  of  the  ellipse  (i.e.,  about 
an  axis  normal  to  its  plane  and  passing  through  its  centre),  so 
that 

7tab(a*  -f  b2}  . 
*~  4 

then: 

M  =  Ga-^  •     •     •     (39) 


5  6  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

Using  /  in  the   manner   shown  in  Eq.  (35),  the  resultant 
shear  at  any  point  becomes,  by  Eq.  (24) : 


T  =  Gar  Vf2  +  2fcos  2^  +  1. 

.      dT  _ 

~d^  - 

gives : 

sin  2(p  =  o,     or     cp  =  90°     or    o°. 

Since  /  is  negative,  T  will  evidently  take  its  maximum 
when  <p  has  such  a  value  that  2f  cos  2cp  is  positive  ;  or,  <p  must 
be  90°. 

Hence  the  greatest  intensity  of  shear  will  be  found  some- 
where along  the  minor  axis.  But  the  preceding  expression 
shows  that  T  varies  directly  as  the  distance  from  the  centre. 
Hence,  the  greatest  intensity  of  shear  is  found  at  the  extremities 
of  the  minor  axis. 

Making  cp  =  90°  and  r  =  b  in  the  value  of  T: 


T=Tm=Gab(i-f)=Ga^~j2.     .     .     (40) 
Taking  Got  from  Eq.  (40)  and  inserting  it  in  Eq.  (38) : 

in  which  : 

nab* 


or  the  moment  of  inertia  of  the  section  about  the  major  axis. 


Art.  10.] 


TORSION  IN  EQUILIBRIUM. 


57 


Equilateral  Triangle  about  its  Centre  of  Gravity. 

This  case  is  that  of  a  cylindrical  piece  whose  normal  cross 
section  is  an  equilateral  triangle,  and  the  torsion  will  be  sup- 
posed about  an  axis  passing  through  the 
centres  of  gravity  of  the  different  nor- 
mal sections.  The  cross  section  is  rep- 
resented in  Fig.  3,  G  being  the  centre 
of  gravity  as  well  as  the  origin  of  co-  H 
ordinates. 

Let    GH  =  y2GD  =  a.     Then    from 
the  known  properties  of  such  a  triangle  : 


Fig.3 


FD  =  DB  =  BF  =  2a 


2a  —  r  cos  cp 
Hence,  the  equation  for  DB  is;  r  sin  cp  - 

Hence,  the  equation  for  BF  is  ;  r  cos  q>  -f-  a  =  o. 

2a  —  r  cos  <p 
Hence,  the  equation  for  FD  is  ;  r  sin  cp  +  •        —-=•         =  O. 

V3 

Taking  the  product  of  these  three  equations,  and  reducing, 
there  will  result  for  the  equation  to  the  perimeter : 


—    —    -r-    COS   "\CP   = 

2         6a  3 


(42) 


Comparing  this  equation  with  Eq.  (33)  : 


Hence: 


$8  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 


sn 


fr*  sin  3<p    , 

—  GaL  —  Ga   —  -.   °^  dr. 
J       60 


since  IP  =  polar  moment  of  inertia  =  3<z4/V/3. 
By  Eq.  (24)  : 


/.     ^—  =  o,     gives     .wz  3^  =  o  ; 

or  <p  =  o°,  60°,  120°,  1 80°,  240°,  300°  or  360 

The  values  o°,  120°,  240°  and  360°  make : 
cos  $(p  —  +  I ; 


And  by  Eqs.  (19)  and  (20)  : 

T.=  -GJ^S.  ......     .     (44) 

T^       Ga(r  -*<%&)     ....    (45) 
Eq.  (22)  then  gives  : 

M  =  Gal,  -  Ga  dr  d<p. 


=  Ga(lp  -    -  ^V^)  =  0.6  Galp  =  1.8  Ga  0VJ;  (46) 


cos  $(p    .      r4  .     N 

_!.+  _.._  (47) 


Art.   10.]  TORSION  IN  EQUILIBRIUM. 


59 


hence,  for  a  given  value  of  r,  these  make  T  a  minimum.    The 
values  60°,  1  80°  and  300°  make: 


cos        = 


hence,    for   a   given   value    of   r  these   make    T  a   maximum. 
Putting  cos  $<p  =•  —  i   in  Eq.  (47)  : 


(48) 


This  value  will  be  the  greatest  possible  when  r  is  the 
greatest.  But  cp  =  60°,  180°  and  300°,  correspond  to  the  nor- 
mal a  dropped  on  each  of  the  three  sides  of  the  triangle  from 
G.  Hence  r  =  a,  in  Eq.  (48),  gives  the  greatest  intensity  of 
shear  Tm9  or  : 


(49) 


Or,  the  greatest  intensity  of  shear  exists  at  the  middle  point  of 
each  side.  Those  points  are  the  nearest  of  all,  in  the  perimeter, 
to  the  axis  of  torsion. 

The  value  of  Ga,  from  Eq.  (49),  inserted  in  Eq.  (46),  gives  : 


in  which  /=  side  of  section  =  2a 


Rectangular  Section  about  an  Axis  passing  through  its  Centre 

of  Gravity. 

In  this   case   it   will  be  necessary  to  consider  one  of  the 
transcendental  forms  to  which  the  integration  of  Eq.  (13)  [or 


60  TORSION  IN  EQUILIBRIUM.  [Art.   IO. 

(26)]  leads  ;  for  if  the  polar  equation  to  the  perimeter  be  formed, 
as  was  done  in  the  preceding  case,  it  will  be  found  to  contain 
r4,  to  which  no  term  in  Eq.  (33)  corresponds. 

If  e  is  the  base  of  the  Napierian  system  of  logarithms  (nu- 
merically, e—  2.71828,  nearly),  and  A  any  constant  whatever, 
it  is  known  that  the  general  integral  of  the  partial  differential 
equation  (13)  may  be  expressed  as  follows  : 


u  =  Aenrcos*  en'rs}n*  ;      .....     (51) 
when  n2  +  n'2  =  o.     For  : 

^L  =  A(n2  +  n' 


dr2         r2  dfy2        r  dr 

But  the  second  member  of  this  equation  is  evidently  equal 
to  zero  if 


(n2  -\-  n'2}  =  o,     or     n'  =  V—  n2. 

These  relations  make  it  necessary  that  either  n  or  ri  shall  be 
imaginary. 

It  will  hereafter  be  convenient  to  use  the  following  notation 
for  hyperbolic  sines,  cosines  and  tangents  : 

sih  t  =  ;  co/i  t  =  — — •  and,  tah  t  =  — -t 


~l 


By  the  use  of  Euler's  exponential  formula,  as  is  well  known, 
and  remembering  that  ri*  =  —  n2,  Eq.  (51)  may  be  put  in  the 
following  form  : 


u  =  ^enrcos^  [An  sin  (nr  sin  qj)  -\-  A'n  cos  (nr  sin  (p]\  ; 
in  which  the  sign  of  summation  is  to  be  extended  to  all  pos- 


Art.  10.]  TORSION1  IN  EQUILIBRIUM.  6  1 

sible  values  of  An  and  An.  At  the  centre  of  any  section  for 
which  r  is  zero,  u  must  be  zero  also,  for  the  axis  of  the  piece  is 
not  shortened.  This  condition  requires  that  A'n  =  O]  u  then 
becomes  : 


u  =  ^enrcos*  AHsin  (nr  sin  cp). 

The  subsequent  analysis  will  be  simplified  by  introducing 
the  form  of  the  hyperbolic  sine,  and  this  may  be  done  by 
adding  and  subtracting  the  same  quantity  to  that  already 
under  the  sign  of  summation,  in  such  a  manner  that  : 

u  =  2[AH  sin  (nr  sin  cp)  .  sih  (nr  cos  cp) 

-\-y2Ansin(nrsincp)e-nrcos*~\    ....     (52) 

Now  if  the  product  : 

sin  (nr  sin  cp)  e~nr  cos  * 

be  developed  in  a  series  and  multiplied  by  AH,  one  term  will 
consist  of  the  quantity  : 

—  r3  sin  cp  cos  cp 

multiplied  by  a  constant,  and  if  : 

n  sin  (nr  sin  cp)  e~nrcos^ 


be  replaced  by  simply  : 

—  at*  sin  cp  cos  cp 

all  the  conditions  of  the  problem  will  be  found  to  be  satisfied. 
This  is  equivalent  to  putting  : 

—  ar*  sin  cp  cos  cp 


62 


TORSION  IN  EQUILIBRIUM. 


[Art.  10. 


for  a  general  function  of  r  sin  cp  and  r  cos  cp.     This  change  will 
give  the  following  form  to  u,  first  used  by  Saint-Venant  : 


u  = 


sin  (nr  sin  cp)  .  sih  (nr  cos  cp)  —  ar2  sin  cp  cos  cp  .     (53) 


Fig.  4  represents  the  cross  section  with  C  as  the  origin  of 
co-ordinates,  and  axis.     The  angle  cp  is  measured  positively 




x£ 

x-     » 

CUx-lT  J  

F 

D 

L 

Fig.* 

from  N  toward  CH.  At  the  points  N,  H,  K  and  L,  in  the 
equation  to  the  perimeter,  dr0  will  be  zero.  Hence,  at  those 
points,  by  Eq.  (21)  : 

-r-  =  ~^\An  sin  (nr  sin  cp)  .  n  cos  cp  .  coh  (nr  cos  cp) 

+  An  .  n  sin  cp  .  cos  (nr  sin  cp)  .  sih  (nr  cos  cpj] 

—  2ar  sin  cp  cos  cp  =  o. 

At  the  points  under  consideration  cp  has  the  values  o°,  90°, 
1 80°,  270°  and  360°.     At  the  points  -AT  and  AT,  <p  =  o°  or  180°  ; 

hence,  sin  q>  =  o,  and  both  terms  of  the  second  member  of  - 

ar 

reduce  to  zero,  whatever  may  be  the  value  of  n.  But  at  H  and 
L,  cp  =  90°  and  270°  ;  hence,  sin  cp  —  -f  i  or  —  I  and  cos  cp  —  o. 

In  order  then,  that  ~  =  o  at  H  and  Z,  these  must  obtain : 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  63 

cos  nr  =  cos  (—  nr)  —  o. 
=  c;  and,  KN  -  b  ;  then  : 


COS  —  = 


(54) 


If  the  signification  of  n  be  now  somewhat  changed  so  as  to 
represent  all  possible  whole  numbers  between  o  and  oo,  Eq.  (54) 
will  be  satisfied  by  writing : 


for  #,  in  that  equation  Eq.  (53)  will  then  become  : 

»    .      .    (2n  —  I                 \       .j  f2n  —  I  \ 

u  =  ^An  sin  ( nr  sin  cp\  .  sih  ( •  nr  cos  tpj 

—  ar*  sin  cp  cos  cp (55) 

The  quantity  An  yet  remains  to  be  determined  by  the  aid 
of  Eq.  (21),  which  expresses  the  condition  existing  at  the 
perimeter  of  any  section. 

Now,  for  the  portion  BN  of  the  perimeter  : 

•    b 
r  cos  cp  =  — , 

and  — ^f-  will  be  the  tangent  of  (-  <p)  ;  or, 
r.d<p 

—  — ^°—  =  —  tan  (—  cp)  =  tan  cp. 
Hence,  Eq.  (21)  becomes  : 


64  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 


du 

~dr 


du 

-- \-  ar 

rdy 


=  t&n  q> (56) 


or: 


du  du 

ar  sin  <p  =  —  cos  q>  ---  =—  sin 
dr  r  dq> 


Substituting  from  Eq.  (55),  then  making  : 


b 

r  co  s  <p  =  —  : 


«   .      2n  —  \  ,  /2n  — 

r  sin  g>  =  2>An  .  -      —  7t  .  coh  ( 

2ac  \      2C 


i     A 
no  ) 

J 

(2n  -  I  \ 

.  sin  (  -  nr  sin  cp  J  . 


If  r  sin  cp  be  represented  by  the  rectangular  co-ordinate 
y,  and  another  quantity  by  //,  the  above  equation  may  be 
written  : 


sm 


y  =  ff,  sin        +  If,  sin 


rr       .     (2n  —   I 

.  .  .  -  Hn  sin    —  -  —  7t     y 


If  both  sides  of  this  equation  be  multiplied  by 

.    f2n  —  i        \       , 
in  r — ny  J  .  dy, 


sin 


Art.   10.]  TORSION-  IN  EQUILIBRIUM.  65 

and  if  the  integral  then  be  taken  between  the  limits  o  and  -   it 

2' 

is  known  from  the  integral  calculus  that  all  terms  except  the 
nth  will  disappear,  and  that : 

c 
rr  f2  .        (211    -     I  \  , 

Hn  =   \   y .  sin   (  -         -  ny  )  .  dy 

Jo  \         6  / 

c 

f~2        .  /2H     —      I  \ 

-f.       sin2  f ny  \  .  dy. 

Completing  these  simple  integrations  : 

H   --  (      _JE \(      i>-«   4 

H«-   \(2n-i}n)   (  T 

Hence : 


.        ,  4  2ac 

(2n  —  i)2  n3  '  c   '  (20  —  i)  ^r 


2C 


If  this  value  of  AM  be  put  in  Eq.  (55),  and  if  rectangular  co- 
ordinates : 

y  =  r  sin  (p,       and   z  = 
be  introduced,  that  equation  will  become  : 


u  =  —  oczy  -\- 

l^~  *y 


-    (57) 


66 


TORSION  IN  EQUILIBRIUM.  [Art.   IO. 

This  value  of  u  placed  in  Eq.  (31)  will  enable  the  moment 
of  torsion  to  be  at  once  written. 

The  limits  -\-y0  and  —  y0  are  -\ —  and ;  and  the  limits 

2  2 


ZQ  and  —  z0  are  -\ —  and  —  -.     Hence  : 


[•1;  - 


.,     2n  —  I 

2  Slk  (  -  -    TtZ 

V      c 


=  (2,  for  brevity. 


2/2  —  1      ,  \       .    f2n  —  I        \ 
nb  }  .  sin  ( ny  j 


(2n  -  if  coh  (— nb 

\     20 


r> 


For  the  next  integration  : 


272—1       ,  4C2  .,    /2H—I       , 

nb  —  -r-      .^   , sih    -      -  nb 

2C 


f  N        .   f2n—\     , 

(2n—\f  coh  (  -      -  nb 

\     2C 


Art.  10.]  TORSION  IN  EQUILIBRIUM. 


67 


r+* 

I   \ 


4C2  ..   /2«  —  I       , 

\7tJ   b    !     fm-° 


Thus  the  integrations  indicated  in  Eq.  (31)  are  completed, 
Hence  : 


Ry  dy  +  alp   -  . 


Remembering  that : 


/2«  —    I 

(-          - 
\     2<: 


.  —  , 

tah    -          -Tib 


.    .    (58) 


But  it  is  known  that : 


x  (2»  _  i)4    -  •    1.2.3*    25 

Hence  Eq.  (58)  becomes  : 


68  TORSION  IN  EQUILIBRIUM.  [Art.   IO. 


M_GabXi      '  64  c  %ta!< 

,  (2«-  i  T/,\n 

fc<v* 

Since : 


/ 1  —  tah  TT         i  —  ta /;  3^,     i  —  tah  5  TT 
"  \         I  ~^"  ~^~ 


e  I  f  I       •    •    •      » 

and  since: 

-^  =  0.209137, 

and  remembering  that : 


295.1215 
Eq.  (59)  becomes : 


M  =  £*&«        -  0.210083    - 


i -tah-          i-tak^  \ 

0.209137-1-    -TJL+        ~^~     -+•••)        (60) 


Eq.  (60)  gives  the  value  of  the  moment  of  torsion  of  a  rec- 
tangular bar  of  material. 


Art.   10.]  TORSION  IN  EQUILIBRIUM.  69 

If  z  had  been  taken  parallel  to  b,  and  y  parallel  to  c,  a 
moment  of  equal  value  would  have  been  found,  which  can  be 
at  once  written  from  Eq.  (60)  by  writing  b  for  c  and  c  for  b. 

That  moment  will  be  : 


M  —  Gacfc  I 0.210083  - 

L3  ** 


,     7TC  , 

I  —  tah  — r         I  —  tah 


T        I  A.  rftVrV  -  J.  •'M'/*'  . 

b  2b    ,  2b 

+  0.209137,1-—  -+    -^r 


.    (61) 


Eq.  (60)  sliould  be  used  when  b  is  greater  than  c,  and  Eq.  (61) 
when  c  is  greater  than  b,  because  the  series  in  the  parentheses 
are  then  very  rapidly  converging,  and  not  diverging.  It  will 
never  be  necessary  to  take  more  than  three  or  four  terms  and 
one,  only,  will  ordinarily  be  sufficient.  The  following  are  the 
values  of, 


for  a  few  values  of  n  : 


_  tah  ~]  =  0.083  :  0.00373  :  0.000162  :  0.000007 
n  =     i  2  3  4 


Square  Section. 
\ic-b  either  Eq.  (60)  or  Eq.  (61)  gives : 

M  —  Gab*         -  0.2101  -f-  0.209/1  -  tah 


7°  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 


.'.    M  =  0.1406  Gab*  =  Got  •  (62) 

42.7  // 


in  which  y2  is  the  area  (  =  $2)  and  7^  is  the  polar  moment  of 

.     /      fr\ 
inertia    =       . 


Rectangle  in  which  b  =  2<r. 
•     If  £  =  2^,  Eq.  (60)  gives  : 


=  (zor  .  2<;4  f-    -  0.105  +  0.1046  (i  —  tah  7i) 


.-.  J/  =  0.457  <^^4  =  <^«  -       ;  •    (6s) 

42  IP 
in  which  ^4  is  the  area  (=  2<2)  and  Ip-=  polar  moment  of  inertia 


Rectangle  in  which  b  —  ^c. 
If  ^  =  ^c,  Eq.  (60)  then  gives  : 

M  =  Gabc*  (-    -  0.0525^  =  1.123 

.-.   M=  Ga-^-—;  .     .     (64) 

40.2  Ip 

in  which  A  =  area  =  ^  and  /  =  polar  moment  of  inertia 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  71 

b&  -f-  I  V 


12  3 

If  £  is  greater  than  2r,  it  will  be  sufficiently  near  for  all 
ordinary  purposes  to  write  : 


1-0.63  .....     (65) 


Greatest  Intensity  of  Shear. 

There  yet  remains  to  be  determined  the  greatest  intensity 
of  shear  at  any  point  in  a  section,  and  in  searching  for  this 
quantity  it  will  be  convenient  to  use  Eqs.  (28)  and  (29). 

It  will  also  be  well  to  observe  that  by  changing  z  to  j,  y 
to  —  z,  c  to  b  and  b  to  c,  in  Eq.  (57),  there  may  be  at  once 
written : 


/2V 

u  =  ay  -  (-) 


—  I         \       .,     2n  —  I 


2;,  _  , 
(2n  -  !)s  ^  (^—  ^—  nc  \ 


This  amounts  to  turning  the  co-ordinate  axes  90' 
Since  the  resultant  shear  at  any  point  is: 


T=  V7?+  7?, 

it  will  be  necessary  to  seek  the  maximum  of 
du 


72  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

The  two  following  equations  will  then  give  the  points  de- 
sired : 


&  fdu 

=       + 


ZL2N 

~&J         fdu  \   f  d*u 


dz  \dy  J  \dzdy 

du 


//;ON 

(68) 


It  is  unnecessary  to  reproduce  the  complete  substitutions  in 
these  two  equations,  but  such  operations  show  that  the  points 
of  maximum  values  of  T  arc  at  the  middle  points  of  the  sides  of 
the  rectangular  sections  ;  omitting  the  evident  fact  that  T  —  o 
at  the  centre.  It  will  also  be  found  that  the  greatest  in- 
tensity of  shear  will  exist  at  the  middle  points  of  the  greater 
sides. 

This  result  may  be  reached  independent  of  any  analytical 
test,  by  bearing  in  mind  that  an  elongated  ellipse  closely  ap- 
proximates a  rectangular  section,  and  it  has  already  been 
shown  that  the  greatest  intensity  in  an  elliptical  section  is 
found  at  the  extremities  of  the  smaller  axis. 

By  the  aid  of  Eqs.  (28),  (29),  (57)  and  (66),  it  will  also  be 
found  that  T3  =  o  at  the  extremities  of  the  diameter  c,  and 
T2  —  o  at  the  extremities  of  the  diameter  b.  The  maximum 
value  of  T  will  then  be  : 


UNIVERSITY 

Art,  10.]  TORSION  IN 


du 

•  (69) 


By  the  use  of  Eq.  (57) : 
du 


Putting  z  =  o  and  y  =  —  in  this  equation,  there  will  result 


y  

-*-    1tt      "~~" 


If  <5  is  greater  than  ^  the  series  appearing  in  this  equation  is 
very  rapidly  convergent,  and  it  will  never  be  necessary  to  use 
more  than  two  or  three  terms  if  the  section  is  square,  and  if  b 
is  four  or  five  times  c  there  may  be  written : 

Tm  =  Gac  •     •    (70 


111 


Square  Section. 

Making  b  —  c  in  Eq.  (70)  and  making  n  -  I,  2  and  3  (i.e., 
taking  three  terms  of  the  series)  there  will  result  : 


74  TORSION  IN  EQUILIBRIUM.  [Art.   IO. 

Tm  =  0.676  Gac     .'.      Ga  =  1.48^. 

Inserting  this  value  in  Eq.  (62)  : 

....     (72) 


in  which  : 

_       &*  be 

I  —  —     and     a  —  —  =  —  . 

12  22 

Rectangular  Section ;  b  =  2c. 

Making  b  —  2c  in  Eq.  (70)   and   making  n  —  I,  only,  there 
will  result  : 

Tm  ~  0.93  Gac    /,     Ga  =  i  .08  — -  . 
Inserting  this  value  in  Eq.  (63) : 

M  =  0.49  ^rw  =  1.47  — ?     .      .     .     .     (74) 


,.     7^0.68^  =  2^;     ....     (75) 


in  which  : 


T       be*         c*  c 

I  —  — -  —  -r     and     a  =  —  . 

12  6  2 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  75 


Rectangular  Section  ;  b  =  4^. 
Making  b  =  ^c  in  Eq.  (70)  and  making  n  =  i,  only  : 

Tm  ~  0.997  Gac     .'.      Ga  =  1.003  —  • 
Inserting  this  value  in  Eq.  (64)  : 

M=  1.126  <*Tm  =  1.69^?     .  (76) 


^M  M 

.6  —  ^  =  0.9  —  ;    .    .    .    .    (77) 


in  which : 


be*         c*  c 

—  = ,—     and     a  =  — 

J2.          3  2 


Circular  Section  about  its  Centre. 

The  torsion  of  a  circular  cylinder  furnishes  the  simplest 
example  of  all. 

If  r0  is  the  radius  of  the  circular  section,  the  polar  equation 
of  that  section  is  : 


^-  =   C,     (constant). 


Comparing  this  equation  with  Eq.  (33),  it  is  seen  that : 


76  TORSION  IN  EQUILIBRIUM.  [Art.  IO. 

By   Eq.  (32)  this  gives  u  =  o.     Hence,  all  sections  remain 
plane  during  torsion. 

Eqs.  (19)  and  (20)  then  give  : 

Txr  =  o ;     and,     7^  =  Gar     ....     (78) 
Eq.  (23)  gives  for  the  moment  of  torsion  : 

M=GaIt (79) 

or : 


M  =  0.5  nr*  .  Ga  =-a (80) 

471  lp 

In  which  equation,  A  is  the  area  of  the  section  and 


I  —  —A 

p  2 

The  greatest  intensity  of  shear  in  the  section  will  be  ob- 
tained by  making  r  —  r0  in  Eq.  (78) ;  or : 

Tm  =  Gar0     .'.     Ga  =  ^ (8l) 

Eq.  (80)  then  becomes  : 

'-.  =  2^- (82) 


~  ,.    M  M  /0  , 

.-.      rw  =  0.64  —  =  0.5  -r  r0 ;      ....     (83) 

'o  * 


in  which 

/=  - 

4 


I  —  — r« 


Art.  10.]  TORSION  IN  EQUILIBRIUM.  77 

It   is  thus   seen  that   the   circular  section  is  the  only  one 
treated  which  remains  plane  during  torsion. 


General  Observations. 

The  preceding  examples  will  sufficiently  exemplify  the 
method  to  be  followed  in  any  case.  Some  general  conclusions, 
however,  may  be  drawn  from  a  consideration  of  Eq.  (33). 

If  the  perimeter  is  symmetrical  about  the  line  from  which 
(p  is  measured,  then  r  must  be  the  same  for  -f  cp  and  —  y, 
hence : 


If  the  perimeter  is  symmetrical  about  a  line  at  right  angles 
to  the  zero  position  of  r,  then  r  must  be  the  same  for : 

cp  —  90°  -|-  qj'     and     90°  —  <p' ; 
hence : 

cl  =  cs  -  cs  .  .  .     =  c't  -  c\  -  c'6  =  .  .  .     =0. 

In  connection  with  the  first  of  these  sets  of  results,  Eq.  (32) 
shows  that  every  axis  of  symmetry -of  sections  represented  by  Eq. 
(33)  will  not  be  moved  from  its  original  position  by  torsion. 

If  the  section  has  two  axes  of  symmetry  passing  through 
the  origin  of  co-ordinates,  then  will  all  the  above  constants  be 
zero,  and  its  equation  will  become : 

—  -j-  c2r2  cos  2cp  -\-  c4r4  cos  4q>  +  c6r6  cos  6cp  -f  .  .  . 


TOR  SIGNAL   OSCILLATIONS.  [Art.  II. 


Art.  ii. — Torsional   Oscillations   of  Circular   Cylinders. 

Two  cases  of  torsional  oscillations  will  be  considered,  in  the 
first  of  which  the  cylindrical  body  twisted  is  supposed  to  be 
the  only  one  in  motion.  In  the  second  case,  however,  the  mass 
of  the  twisted  body  will  be  neglected,  and  the  motion  of  a 
heavy  body,  attached  to  its  free  end,  will  be  considered.  In 
both  cases  the  section  of  the  cylinder  will  be  considered  cir- 
cular. 

Since  these  cases  are  those  of  motion,  the  internal  stresses 
are  not,  in  general,  in  equilibrium ;  hence,  equations  of  motion 
must  be  used,  and  those  of  Art.  7  are  most  convenient.  Of 
these  last,  the  investigations  of  the  preceding  Art.  show  that 
Eq.  (4)  is  the  only  one  which  gives  any  conditions  of  motion  in 
the  problem  under  consideration. 

Putting  the  value  of  : 

^dw 


in  Eq.  (4)  of  Art.  7,  that  equation  may  take  the  form  : 
G 


Y* 

For  brevity,  b*  is  written  for  —  . 

That  dimension  of  the  cross  section  of  the  body  which  lies 
in  the  direction  of  the  radius  will  be  assumed  so  small  that  w 
may  be  considered  a  function  of  x  and  /  only.  The  results  will 
then  apply  to  small  solid  cylinders  and  all  hollow  ones  with 
thin  walls. 

The  general  integral  of  Eq.  (i),  on  the  assumption  just 
made,  is  (Books'  "  Differential  Equations,"  Chap.  XV.,  Ex.  i) : 


Art.  II.]  CIRCULAR   CYLINDERS.  /Q 

w  =  /(*  +  ^)  -r  F(x  -  bt)\ 

in  which  /  and  F  signify  any  arbitrary  functions  whatever. 
Now  it  is  evident  that  all  oscillations  are  of  a  periodic  char- 
acter, i.e.,  at  the  end  of  certain  equal  intervals  of  time,  w  will 
have  the  same  value.  Hence  since /"and  -Fare  arbitrary  forms, 
and  since  circular  functions  are  periodic,  there  may  be  written: 


w  =  An\sin  (pinx  -f-  anbf)  +  jzVz  (ovr  —  < 

—  Bn\cos  (cxnx  -j-  (*nbt)  —  cos  (i*nx  —  a'nb£)\  ;  .     .     (2) 

in  which  an,  An  and  BH  are  coefficients  to  be  determined. 

Substituting  for  the  sines  and  cosines  of  sums  and  differ- 
ences of  angles : 

w  =  2  sin  anx  (An  cos  anbt  -j-  Bn  sin  anbf)  ...     (3) 

Let  the  origin  of  co-ordinates  be  taken  at  the  fixed  end  of 
the  piece ;  w  must  then  be  equal  to  zero,  as  is  shown  by  Eq. 
(3).  But  there  may  be  other  points  at  which  w  is  always  equal 
to  zero,  whatever  value  the  time  t  may  have.  These  points, 
called  nodes,  found  by  putting  w  =  o  ;  or : 

sin  ax  =  o (4) 

This  equation  is  satisfied  by  taking: 

it     2ir.      3?r  nn . 

ex..  ^^      ,          ,    •      »•••>        > 
a-  a       a  a 

and  x  =  a ;  in  which  a  is  the  length  of  the  piece.      , 
Hence,  at  the  distances  : 


SO  TOR  SIGNAL    OSCILLATIONS.  [Art.  II. 


a      a  a 

"'   2  '    3  '  '     '  '  n 

from  the  fixed  end  of  the  piece,  there  will  exist  sections  which  are 
never  distorted  or  moved  from  their  positions  of  rest.  These  are 
called  nodes,  and  one  is  assumed  at  the  free  end,  although  such 
an  assumption  is  not  necessary,  since  a  is  really  the  distance 
from  the  fixed  end  to  the  farthest  node  and  not  necessarily  to 
the  free  end. 

If,  as  is  permissible,  An  and  Bn  be  written  for  twice  those 
quantities,  the  general  value  of  w  now  becomes : 

TTX  (  .  nbt          „      .      nbt  \ 

w  =  sin A,  cos +  B,  sin  • — 

a    \    '  a  a   J 

.     2nx  f  ,          2nbt         „      .    2nbt\ 

•+•  sin A~  cos 4-  />,  sm  • —  -  ) 

a    \    '  a  a   ) 

$nbt 


.  . 

sin  -  —  A,  cos 


^7rbt\ 

-f  g.  sin  -  --  ) 
a   J 


.                 .           nnbt         n      .     nnbt  ,  N 

sin  --(An  cos  — h  BK  sin  —-        -     .     ($) 


The   coefficients  A   and  B   are   to  be  determined    by  the 
ordinary  procedure  for  such  cases,     Let  : 


be  the  expression  for  the  initial  or  known  strain  at  any  point, 
for  which  the  time  t  is  zero.  Then  if  An  is  any  one  of  the  co- 
efficients A  : 


Art-  I*-]  CIRCULAR    CYLINDERS.  8 1 


* (6) 

The  velocity  at  any  point,  or  at  any  time,  will  be  given  by : 

dw  .    nx (  .      .     nbt  nbt\    nb 

— _  =  —  sin  —  (A^sin  ~ B.cos-   -  )  —  .          (7) 

dt  a\  a  a  J    a  {/> 

In  the  initial  condition,  when  the  time  is  zero,  or  /  =  o,  it 
has  the  given,  or  known,  value : 


Then,  as  before  : 


Thus  the  most   general   value   of  w  is  completely  deter- 
mined. 

The  intensity  of  shear  at  any  place  or  time  is  given  by: 


w  being  taken  from  Eq.  (5). 

The  second  case  to  be  treated  is  that  of  the  torsion  pen- 
dulum, in  which  the  mass  of  the  twisted  body  is  so  incon- 
siderable in  comparison  with  that  of  the  heavy  body,  or  bob, 
attached  to  its  free  end  that  it  may  be  neglected. 
6 


82  TOR  SIGNAL   OSCILLATIONS.  [Art.  II. 

Let  M  represent  the  mass  of  the  pendulum  bob,  and  k,  its 
radius  of  gyration  in  reference  to  the  axis  about  which  it  is  to 
vibrate  ;  then  will  MJ?  be  its  moment  of  inertia  about  the 
same  axis. 

The  unbalanced  moment  of  torsion,  with  the  angle  of 
torsion  a,  is,  by  Eq.  (9)  of  Art.  10: 


The  elementary  quantity  of  work  performed  by  this  un- 
balanced couple,  if  fi  is  the  general  expression  for  the  angular 
velocity  of  the  vibrating  body,  is  : 

Galp  .  /3  dt. 

This  quantity  of  energy  is  equal  in  amount  but  opposite  in 
sign  to  the  indefinitely  small  variation  of  actual  energy  in  the 
bob  ;  hence  : 


Galpp  dt  =  -  d  (~^-)  =  ~  Mfrfi  dp. 
But  if  a  is  the  length  of  the  piece  twisted  : 


,    and    dfi  = 


, 

at  at 


Multiplying  this  equation  by  2d(aa),  and  for  brevity  put- 
ting: 


Art.  II.]  TORSION  PENDULUM.  83 

then  integrating  and  dropping  the  common  factor  a*  : 


When  a  —  av  the  value  of  the  angle  of  torsion  at  the  ex- 
tremity of  an  oscillation,  the  bob  will  come  to  rest  and  -r-  will 
be  zero.  Hence  : 

C  =  //*,', 
and 


da                     I~H     j. 
—  =   ,  / .  at. 


+  (<:'  =  o).     ...     (9) 


C'  =  o  because  o'  and  t  can  be  put  equal  to  zero  together. 
At  the  opposite  extremities  of  a  complete  oscillation  a  will 
have  the  values  : 

(+  «,)     and     (-  *,). 
Putting  these  values  in  the  expression  : 


and  taking  the  difference  between  the  results  thus  obtained, 


84  THICK,  HOLLOW  SPHERES.  [Art.  12. 

the  following  interval  of  time  for  a  complete  oscillation  will  be 
found  : 


t  =  n 


The  time  required  for  an  oscillation  is  thus  seen  to  vary 
directly  as  the  square  root  of  the  moment  of  inertia  of  the  bob  and 
the  length  of  the  piece ,  and  inversely  as  the  square  root  of.  the  co- 
efficient of  elasticity  for  shearing  and  the  polar  moment  of  inertia 
of  the  normal  section  of  the  piece  twisted. 

The  number  of  complete  oscillations  per  second  is  — .     If 

this  number  is  the  observed  quantity,  the  following  equation 
will  give  G : 


It 


The  formulas  for  this  case  should  only  be  used  when  the 
mass  of  the  cylindrical  piece  twisted  is  exceedingly  small  in 
comparison  with  M. 


Art.  12. — Thick,  Hollow  Spheres. 

In  order  to  investigate  the  conditions  of  equilibrium  of 
stress  at  any  point  within  the  material  which  forms  a  thick 
hollow  sphere,  it  will  be  most  convenient  to  use  the  equations 
of  Art.  8.  As  in  the  case  of  a  thick,  hollow  cylinder,  the  in- 
terior and  exterior  surfaces  of  the  sphere  are  supposed  to  be 
subjected  to  fluid  pressure. 

Let  r'  and  r,  be  the  interior  and  exterior  radii,  respect- 
ively. 


Art.  12.]  THICK,  HOLLOW  SPHERES.  85 

Let  —  /  and  —  /,  be  the  interior  and  exterior  intensities, 
respectively. 

Since  each  surface  is  subjected  to  normal  pressure  of  uni- 
form intensity  no  tangential  internal  stress  can  exist,  but  normal 
stresses  in  three  rectangular  co-ordinate  directions  may  and  do 
exist.  Consequently,  in  the  notation  of  Art.  8, 


=  o. 


With  a  given  value  of  r,  also,  a  uniform  state  of  stress  will 
exist.  Neither  N^  nor  N$  can,  then,  vary  with  <p  or  ip.  By  the 
aid  of  these  considerations,  and  after  omitting  7?0,  #0,  Ww  and 
the  second  members,  the  Eqs.  (i),  (2)  and  (3)  of  Art.  8  reduce 
to: 


_    +    2Nr  -N,-N,    =  0 


-  Ai,   +  ^    _  o       ......      (2) 

ByEq.(2): 

Nt±N+. 

Eq.  (i)  then  becomes  : 


On  account  of  the  existing  condition  of  stress,  which  has 
just  been  indicated,  it  at  once  results  that  : 

rj  =  oo  =  o, 
and  that  p  is  a  function  of  r  only. 


86  THICK,  HOLLOW  SPHERES.  [Art.  12. 

Eqs.  (4)  to  (10),  of  Art.  8,  then  reduce  to  : 

«=!+-,? » 


.    .    .    .    (5) 
.    (6) 


i  —  21  r 

After  substitution  of  these  quantities,  Eq.  (3)  becomes : 
2rdp-  2pdr\    ,    ^rd*f>  dp 


or : 


df  ^      dr       =  °" 


One  integration  gives : 


Hence  #,  the  rate  of  variation   of  volume,  is  a  constant 
quantity.     Eq.  (7)  may  take  the  form  : 

r  dp  -f-  2p  dr  =  cr  dr. 


Art.  12.]  THICK,  HOLLOW  SPHERES.  87 

As  it  stands,  this  equation  is  not  integrable,  but,  by  inspect- 
ing its  forro,  it  is  seen  that  r  is  an  integrating  factor.  Multi- 
plying both  sides  of  the  equation,  then,  by  r  : 

r*  dp  +  2rp  dr  —  d(r*p)  =  cr*  dr. 


Substituting  from  Eqs.  (7)  and  (8)  in  Eq.  (5)  : 


It  is  obvious  what  ^  represents. 

When  /  andrx  are  put  for  r,  Nr  becomes   -/  and    -  pv 


Hence: 


and  : 


* 

--= -A 


These  equations  express  the  conditions  involved  in  Eqs. 
(13),  (14)  and  (i  5),  of  Art.  6. 
The  last  equations  give: 


88  THICK,  HOLLOW  SPHERES.  [Art.  12. 


These  quantities  make  it  possible  to  express  Nr  and  N$  in- 
dependently of  the  constants  of  integration,  c  and  b,  for  those 
intensities  become  : 


N    =  /vjl  -_J^__  _  \r*      f)'  '_  .    i.    .     (lo) 

^'3    ^-3  f3    ^3  ^-3 


^  _  ^    _  tS?-pr*       te^ 

/s  _.  ris  2(r's  _  rf)          r* 

Thus  it  is  seen  that  N$  =  N+  has  its  greatest  value  for  the 
interior  surface  ;  that  intensity  will  be  called  k. 

It  is  now  required  to  find  r^  —  rr  =  t  in  terms  of  hy  p 
and  /j. 

If  r=  /in  Eq.  (11): 


Dividing  this  equation  by  r'3  and  solving  : 

^_         2(7* 
r'3  "  2k'- 


If  the  intensities  /  and/,  are  given  for  any  case,  Eq.  (12) 
will  give  such  a  thickness  that  the  greatest  tension  h  (suppos- 
ing /,  considerably  less  than/)  shall  not  exceed  any  assigned 


Art.   12.]  -THICK,  HOLLOW  SPHERES.  89 

value.  If  the  external  pressure  is  very  small  compared  with 
the  internal,  /x  may  be  omitted. 

The  values  of  A  and  ^Gb  allow  the  expressions  for  c  and  b 
to  be  at  once  written. 

If  A  is  greater  than  /,  nothing  is  changed  except  that 
N$  =  N$  becomes  negative,  or  compression. 


CHAPTER  III. 

THE  ENERGY  OF  ELASTICITY. 

Art.  13. — Work  Expended  in  Producing  Strains. 

THE  general  expressions,  in  rectangular  co-ordinates,  for  the 
unbalanced  forces  which  act  in  the  three  co-ordinate  directions 
upon  any  indefinitely  small  parallelopiped  of  material  subjected 
to  any  state  of  stress  whatever,  are  given  by  multiplying  each 
of  Eqs.  (7),  (8)  and  (9)  of  Art.  6  by  (dx  dy  dz).  If  an  indefi- 
nitely small  change  in  the  state  of  stress  takes  place,  that  in- 
definitely small  parallelopiped  will  suffer  a  displacement  whose 
rectangular  components  are  du,  dv,  dw ;  and  the  amount  of 
work  performed  in  moving  it  will  be  found  by  multiplying  each 
of  the  three  unbalanced  forces,  determined  as  above,  by  each 
of  the  three  small  strains  belonging  to  the  same  direction  with 
the  force  (as  in  Art.  6,  u,  v  and  w  are  strains  in  the  directions 
of  x,  y  and  s).  This  differential  quantity  of  work,  integrated 
throughout  the  extent  of  the  body,  will  give  the  elementary 
quantity  of  work  required  for  the  small  deformation  and  mo- 
tion of  the  whole  body. 

The  resulting  equations  form  the  foundation  of  investiga- 
tions in  elastic  vibrations  and  resilience  ;  they  also  furnish  the 
means  of  reaching  some  general  conclusions  in  reference  to 
suddenly  applied  loads. 

Let  dW  represent  the  elementary  quantity  of  work  required 
for  the  motion  only,  then  the  operations  which  have  just  been 
indicated  will  give  the  following  expression  : 


Art.  13.]      WORK  EXPENDED  IN  PRODUCING  STRAINS. 

JIIKsr  dx  d*  *  +     d*  *  *  +     **  dy  d 


+ 


dx  *&"**  +      dx  d  dz  +  ~  dx 


du  +  Y,dv  +  Z.  dw^  dx  dyd^\  = 


This  equation,  however,  can  be  put  in  a  much  simpler  form, 
and,  caused  to  take  a  shape  which  will  show  at  a  glance  the 
true  character  of  each  part  ;  dx,  dy  and  dz  are  differentials  of 
independent  variables,  hence  they  are  arbitrary  and  independ- 
ent. Integrating  by  parts,  therefore  : 


dx  - 


in  which  the  primes  indicate  the  values  of  N^  and  u  at  one 
point  of  the  exterior  surface  of  the  body,  and  the  seconds  those 
values  for  another  point  of  the  exterior  surface ;  these  points 
being  taken  at  opposite  extremities  of  #  bar  of  the  material 
whose  normal  section  is  (dy  dz)  and  which  extends  entirely  through 


92  THE  ENERGY  OF  ELASTICITY.  [Art.  13. 

the  body  in  the  direction  of  x.     Maintaining  the  same  notation 
and  proceeding : 

~~  dy  .  dx  dz  .  du  =    1(7^'  du1  —  T^'  du"}  dx  dz 
du 


-ji  ds  .dxdy.du=  [  J(7V  du'  -  T,"  du'}  dx  dy 


But  by  referring  to  the  equations  which  immediately  pre- 
cede (13),  (14)  and  (15)  of  Art.  6,  it  will  be  seen  that  the  sum 
of  these  three  double  integrals  will  represent  the  amount  of  work 
performed  on  the  body  by  the  external  forces  acting  in  the  direc- 
tion of  the  axis  of  x.  Precisely  the  same  general  results  are 
obtained  for  the  directions  of  y  and  z  by  treating  in  the  same 
manner  the  remaining  derivatives  of  the  internal  intensities  in 
Eq.  (i).  The  preceding  operations  are  typical,  therefore  they 
need  not  be  repeated. 

Again,  by  reference  to  the  notation  and  demonstrations  of 
Art.  5  : 


du 


Art.  13.]       WORK  EXPENDED  IN  PRODUCING  STRAINS. 


93 


.  Finally  : 


^§=1'(§)'- 

Introducing  these  reductions  and    quantities,  Eq.  (i)  be- 
comes : 

\P'da'  (cos  it'  du'  -f  cos  x'  dv'  -f  cos  p'  dw) 
YF'da"  (cos  n"  du"  +  cos  x"  dv"  +  cos  p"  dw"} 

•f  F0  dv  +  ZQ  dw)  dx  dy  dz 


Eq.  (2)  shows  clearly  the  distribution  of  the  different  por- 
tions of  work  expended.  The  first  two  (single)  integrals  evi- 
dently represent  the  total  amount  of  work  performed  by  forces 
acting  on  the  exterior  surface  of  the  body ;  it  will  be  indicated 
by  dW^.  If  the  forces  P'  and  P"  are  of  the  same  kind  (*>., 
both  pulls  or  both  pushes),  the  algebraic  sum  of  any  two  terms 


, 

94  THE  ENERGY  OF  ELASTICITY.  [Art.  13. 

of  these  integrals  will  be  a  numerical  sum  if  they  involve  co- 
sines of  the  same  letter  but  of  opposite  signs. 

The  correct  application  of  Eq.  (2)  depends  largely  upon  the 
proper  observance  of  the  signs  which  should  affect  P',  P"  and 
the  cosines. 

The  first  triple  integral  in  the  first  member  of  Eq.  (2),  in 
which  each  intensity  of  stress  is  multiplied  by  the  differential 
of  its  characteristic  strain,  and  which  will  be  indicated  by  dW2, 
is  evidently  the  amount  of  work  required  for  the  small  distor- 
tion alone,  of  the  body.  The  quantity  within  the  parentheses 
is  called  the  potential  energy  of  the  elasticity  of  a  cubic  unit  of 
material,  since,  if  it  be  multiplied  by  (dx  dy  ds),  the  product  will 
express  the  amount  of  work  that  small  portion  of  material  can 
perform  in  returning  to  its  original  condition. 

This  potential  energy  for  a  cubic  unit  is  easily  integrated 
by  the  aid  of  Eqs.  (u),  (12),  (13),  (17),  (18)  and  (19)  of  Art.  5. 
Making  the  substitutions  from  those  equations  and  integrat- 
ing : 


—   27- 


H  is  the  potential  energy  of  a  cubic  unit  of  material  for  a 
change  of  state  extending  from  the  limit  o  to  the  strains  lv  /2, 
etc. 

The  last  triple  integral  in  the  first  member  of  Eq.  (2)  ex- 
presses the  work  done  by  external  forces  which  take  hold  of 
the  mass  of  the  body.  Let  it  be  represented  by  dWy  This 


UNIVERSITY 
&  >  _e?_ 

Art.  13.]       WORK  EXPENDED  IN  PRODUCING  s] 

triple  integral  added  to  the  first  two  single  integrals,  which 
belong  to  the  surface  of  the  body,  will  give  the  total  work 
done  by  external  forces. 

The  second  member  of  the  equation  is  the  small  variation 
of  actual  energy,  which  usually  exists  in  consequence  of  vibra- 
tions. 

Let  V  be  the  resultant  velocity  of  the  parallelepiped,  then 
will: 


• 
By  transferring  dW^  the  first  two  members  of  Eq.  (2)  may 

take  the  form  : 


dW,  +  dW^  =  dW>+\(\mVdVdx  dy  dz. 

ds    .     .     (3) 


Or,  the  total  external  work  performed  is  equal  to  the  work 
done  in  distorting  the  body  added  to  the  change  of  actual  en- 
ergy. 

This  result  expresses  the  law  of  the  conservation  of  energy 
for  the  elastic  bodies  considered. 

If  the  external  work  is  nothing,  the  first  member  of  Eq.  (3) 
is  zero.  The  actual  energy  will  then  exist  in  consequence  of  a 
state  of  vibration.  Let  its  variable  value  be  represented  by  U. 
Since  dx,  dy,  and  dz  are  arbitrary  : 


96  THE  ENERGY  OF  ELASTICITY.  [Art.  14. 

C  representing  a  constant  of  integration.     Under  the  circum- 
stances assumed,  then  : 

W2+  U=C (4) 

Hence,  the  total  energy  of  the  vibrating  body  (i.e.,  the  sum  of  the 
actual  and  potential}  will  be  constant. 


Art.  14.— Resilience. 

The  term  resilience  is  applied  to  the  quantity  of  work  which 
is  required  to  be  expended  in  order  to  produce  a  given  state  of 
*  strain  in  a  body.  The  analytical  expression  for  this  amount  of 
work  is  obtained  directly  from  Eq.  (2)  of  the  preceding  Art. 

Let  the  simple  case  of  a  single  straight  bar  be  considered  ; 
and  let  all  the  external  forces  act  parallel  to  the  axis  of  the  bar 
while  they  take  hold  of  the  end  surfaces,  which  are  normal 
sections.  These  external  forces  will  be  considered  equal  to  the 
internal  stresses  developed  ;  consequently  no  vibrations  will 
exist.  The  action  of  the  external  forces  XQJ  YQ  and  Z0  will  also 
be  omitted. 

Now,  if  the  axis  of  x  be  taken  parallel  to  the  axis  of  the 
bar,  and  if  that  end  of  the  bar  to  which  P"  is  applied  be  fixed, 
there  will  result  from  the  preceding  conditions : 

COS  7t'  =   COS  7t"  =    I, 

cos  x  =  cos  p'  =  cos  x"  =  cos  p"  =  du"  =  O, 

N,  =  N,  =  T,  =  T,  =  T3  =  a 
Eq.  (2)  will  then  become  : 

(P'da  du'  =  fff^V;  dl,  dx  dy  dz      ....     (i) 


Art.   14.]  RESILIENCE. 


97 


But  if  the  intensity  P'  is  uniform  and  A  the  area  of  normal 
section,  Eq.  (i)  becomes  : 

P'A  du'  =  AN,  x,dl,    ......    (2) 

in  which  x,  is  the  length  of  the  bar. 
From  Eq.  (i)  of  Art  I  : 


hence  : 

\P'A  du'  =  Resilience  =  Ax,  E  [''  /x  dl,  =  A^Z^.     (3) 

J  Jo  2 

The  quantity  : 

TV2 
2  —  —i- 


is  called  the  "Modulus  of  Resilience."     This  term   is   usually 
applied  when  N,  is  the  greatest  intensity  allowed  in  the  bar. 

If  one  end  of  a  bar,  placed  in  a  vertical  position,  is  fixed, 
while  a  falling  body  whose  weight  is  wt  acts  upon  the  other 
end,  the  height  of  fall  may  be  sufficient  to  produce  rupture. 
Let  h  be  the  height  of  fall  required  and  N,  —  p  the  ultimate 
resistance  of  the  material  of  the  bar.  In  order  that  rupture 
may  take  place  : 

Ax.      p*  A       p 

' 


Eq.  (4)  shows  that  the  height  of  fall  varies  directly  as  the 
length  of  the  piece.  It  is  virtually  assumed,  however,  that  the 
extension  or  compression  is  uniform  throughout  the  length  of 

7 


98  THE  ENERGY  OF  ELASTICITY.  [Art.   15. 

the  bar,  to  the  instant  of  rupture.  This,  in  reality,  is  not  true, 
and  h  will  not  vary  as  rapidly  as  xv  The  principle  established 
in  Eq.  (4)  is  equally  true  for  torsion  and  bending. 


Art.  15. — Suddenly  Applied  External   Forces  or  Loads. 

A  very  important  deduction  can  be  reached  by  an  attentive 
consideration  of  Eq.  (2)  of  Art.  13,  if  it  be  assumed  that  the 
external  forces  P'  and  P"  are  simple  and  direct  functions  of 
the  external  strains  uy  v  and  w.  In  such  a  case  the  following 
relations  will  hold,  in  which  a,  b  and  c  are  constants  : 

P'  cos  n'  =  au'  ;     P'  cos  %'  =  bv'  ;     P'  cos  p'    =  cw'  ; 
P"  cos  7t"  =  au"  ;     P"  cos  *"  =  bv"  ;     P"  cos  p"  =  cw"  . 

Consequently  the  external  work  performed,  omitting  Xw  YQ 
and  Zw  in  changing  the  body  from  a  state  of  no  stress  to  that 
indicated  by  the  strains  u'1  t)',  to'-,  u"1  t)"1  to",  will  be  : 


in  which  equations  the  integrals  are  to  be  made  to  cover  the 
whole  extent  of  the  surface. 

If,  instead  of  being  variable,  the  forces  P'  and  P"  are  con- 
stant and  equal  to  the  final  values  of  the  preceding  case  (i.e., 
equal  to  #u',  £o',  OD',  #tt",  etc.),  the  external  work  performed  in 
bringing  the  body  to  the  final  state  u',  t)',  etc.,  will  be : 


Art.  15.]  SUDDENLY  APPLIED  LOADS.  99 

\dW^  =  (da1  (an'2  -\-  bv'2  +  no'2) 

U*"  (an"2  H-  fo"2  +  no"2)  =  2  0". 

This  last  case  is  that  of  "  suddenly  "  applied  external  forces 
or  loads,  while  the  former  is  that  of  gradual  application,  in 
which  the  external  forces,  at  each  instant,  are  equal  to  the  in- 
ternal resistances.  In  the  case  of  sudden  application  it  is  seen 
that  the  amount  of  work  expended  is  twice  as  great  as  in  the 
other  case  ;  consequently  when  the  body  arrives  at  the  state  of 
strain  indicated  by  n',  t>',  etc.,  there  remains  to  be  expended  just 
as  much  work  as  has  already  been  performed,  and  at  the  instant 
in  question  it  exists  in  the  body  in  the  shape  of  actual  energy. 

But  if  an  amount  of  energy  equal  to  W  will  produce  the 
strains  tt',  t)',  etc.,  and  if,  while  the  force  acts  which  performed 
the  work,  an  additional  amount  of  energy  equal  to  W  be  ex- 
pended on  the  body,  additional  strains  equal  to  tl',  t)',  etc.,  will 
be  produced  in  the  body. 

When  the  body  comes  to  rest,  therefore,  the  external  strains 
will  be  21:',  2t)',  2tn',  etc.  There  is  then  no  actual  energy,  all  is 
potential. 

Since  the  external  strains  are  211',  2t)',  etc.,  the  external 
work  which  has  been  performed  up  to  this  instant  will  be  found 
by  putting  those  quantities  in  the  place  of  tt',  fl',  etc.,  in  the 
expression  for  H7',  above.  That  expression  will  then  become 


For   gradually  applied  loads   Eq.   (2)  of  Art.  13  becomes 
simply  : 


W  =  \\\H  dx  dy  dz\ 
in  which  H  is  the  potential  energy  per  cubic  unit  for  the  si:: 


I OO  THE  EN  ERG  Y  OF  EL  A  S  TICIT  Y.  [  A  rt.   1 6 

of  strain  corresponding  to  u',  t)',  to',  etc.  But,  if  the  loads  be 
suddenly  applied,  in  accordance  with  what  has  been  given,  the 
Eq.  (2)  of  Art.  13  becomes  : 


=  (\\4Hdxdydz. 


Now  the  expression  for  H,  given  in  Art.  13,  shows  that 
multiplying  H  by  4  is  the  same  thing  as  doubling  the  strains  : 

4  4  ly  9»  <P2  and  cpy 

But  by  doubling  the  strains  the  intensities  of  stresses  are 
doubled.  Hence,  if  the  same  loads  are  first  applied  gradually 
and  then  suddenly,  the  strains  and  stresses  in  the  latter  case  will 
be  double  those  in  the  former.  This  is  a  very  important  principle 
in  engineering  practice,  for  it  covers  all  cases  of  tension,  com- 
pression, torsion  and  bending.  It  also  finds  many  important 
extensions  in  special  cases  of  such  structures  as  iron  and  steel 
bridges,  particularly  suspension  bridges.  For  the  considera- 
tions involved  in  this  Art.  show  that  in  all  cases  of  sudden 
applications  of  loads,  actual  energy  will  be  stored  and  restored 
during  different  intervals  of  time  and,  consequently,  that  vibra- 
tions will  be  initiated. 

Eq.  (2)  of  Art.  13  furnishes  a  most  convenient  and  elegant 
point  of  departure  for  investigations  in  such  special  cases,  as 
will  be  exemplified  in  the  next  Art*. 


Art.  16. — Longitudinal   Oscillations   of  a  Straight    Bar  of  Uniform 

Section. 

The  complete  solution  of  this  problem  will  not  be  given, 
though  it  may  be  reached. 

Let  the  bar  be  fixed  at  one  end  in  a  vertical  position  and 


Art.   1 6.]  LONGITUDINAL   OSCILLATIONS.  IOI 

let  a  heavy  weight,  W,  act  on  the  other.  Also,  let  the  axis  of 
x  be  taken  parallel  to  the  axis  of  the  bar,  whose  uniform  nor- 
mal section  will  be  represented  by  A. 

On  account  of  the  circumstances  of  application  of  the  ex- 
ternal forces  and  position  of  bar,  the  following  equations  of 
condition  will  exist : 

cos  /  =  cos  p'  =  cos  x"  =  cos  P"  =  du"  =  N2  =  NS 
=  T,  =  Ta  =  T3  =  o  =  F0  =  Z0. 

dv     _  dw 

~dt      ''  ~di 

will   be   very  small   compared   with  -^  ,   hence  they  will   be 

omitted.  P1  is  the  heavy  weight  attached  to  the  free  end  of 
the  bar  divided  by  A  ;  consequently: 

cos  7t'  =  i. 
Eq.  (2)  of  Art.  13,  now  reduces  to  : 

(P'da1  du1  -  ( j  J£/x  dk  dx  dy  dz  +  JJJjro  du  dx  dy  dz 


m  — 


The  integrals  are  to  be  extended  throughout  the  whole  of 
the  bar.     Since  strains  and  stresses  are  uniform  for  any  one 
cross  section   of  the  bar,  and  because  XQ  =  w  =   weight  of  a 
unit  of  volume  of  the  bar  (the  force  of  gravity  is  the  only  < 
ternal  force  which  acts  on  the  mass  of  the  bar),  Eq.   1 


comes : 


102  THE  ENERGY  OF  ELASTICITY.  [Art.  16. 

Wdu'  -  AE  *±-  dx  +  Awu  dx  =  -  A  (^]  dx  +  C dx  .     (2) 

2  2          V#  / 

This  equation  (C  being  a  constant  of  integration)  involves 
the  complete  problem  of  longitudinal  oscillations.  Two  spe- 
cial cases,  only,  however,  will  be  treated,  in  which  the  weight  of 
the  bar  is  so  small  compared  with  W  that  it  may  be  neglected. 
This  condition  involves  the  omission  of  : 

,  ,     m     ,  fdu\2   , 

Awu  dx     and       -A(—}  dx , 
2       \dt) 

in  Eq.  (2),  and  makes  /x  constant  throughout  the  length  of  the 
bar. 

Since  the  equation  must  be  homogeneous,  C  will  represent 
a  quantity  of  actual  energy  ;  in  fact,  a  part  of  that  quantity 
stored,  at  any  instant,  in  W. 

If  x^  represents  the  length  of  the  bar,  C  may  be  put  equal 
to  : 

W    fdu\* 


2gx,  \dt  J 
Also,  because  !t  is  constant  for  the  whole  bar : 


Introducing  all  these  changes  in  Eq.  (2)  and  integrating : 

Tj/  '        AKU'*          w  (du\  , 

Wu  —  AE — •  =  —  ( —      -f  C   .     .     .     .     (3) 

2*,  2g    \dtl 

If  W  is  suddenly  applied  to  the  bar  while  in   a  state  of 
equilibrium  or  rest,  for  which  : 


Art.  1 6.]  LONGITUDINAL   OSCILLATIONS. 


103 


du' 


C'  will  be  zero,  as  the  equation  shows  by  such  a  substitution. 
For  this  case  Eq.  (3)  becomes,  after  omitting  the  primes : 


dt  = 


IWx, 


du 


A  Eg 


/2WxT    , 

VIET*' 


AEu 


ver  sin        -=^ 


f  N. 
..     (4) 


The  limits  of  the  amplitude  are  discovered  by  putting  : 
^-'     (the  velocity)     =  o, 

in  Eq.  (3),  remembering  that  C'  is  also  equal  to  zero.     That 
operation  will  give  : 


u  —  o     and     u  — 


AE    * 


Putting  these  values  in  Eq.  (4),  successively,  and  taking  the 
difference  of  the  results,  the  time  occupied  by  one  oscillation 
will  be  : 


/*| 

ver  sin  -  K  \  I  —     • 

o 


T-       IWX* 
~  \j  AE 

in  which  equation  : 


IO4  THE  ENERGY  OF  ELASTICITY.  [Art.   1  6. 


is  the  strain  in  the  bar  caused  by  a  gradual  application  of  W. 

In  the  second  case  to  be  treated  the  bar  is  first  supposed  to 
take  a  vertical  position,  with  the  weight  attached  to  its  free 
end,  in  a  state  of  equilibrium.  An  external  force  then  de- 
presses the  free  end  a  distance  u0,  measured  from  its  position  of 
equilibrium.  If  the  force  F  is  now  removed,  the  weight  will 
make  excursions  on  each  side  of  its  position  of  rest. 

Let  #,  represent  the  value  of  u'  corresponding  to  the  weight 
W  alone,  as  in  the  previous  case  ;  then  let  : 


u  being   measured    from   the    position   of  equilibrium  of   the 
weight  W. 

Eq.  (3)  will  then  take  the  form  : 


.    (6) 


When  u  =  UQ  the  body  comes  to  rest.     Hence  : 

i 
W(Ul  +  u0)  -  *£-  (u,  +  uJ=C.    .    .    .     (7) 

•**! 

Subtracting  Eq.  (7)  from  Eq.  (6)  : 

W(u  -  O  -  ||  [2«,  (u  -  «0)  +  «"  -  «0>]  =  |J  (  J)°  -   (8) 

since  : 

d(Ui  -}-«)  =  du. 

» 
Remembering  that  : 


Art.   id]  LONGITUDINAL    OSCILLATIONS.  105 


Eq.  (8)  may  take  the  form  : 


dt  = 


du 


(9) 


(10) 


Eq.  (9)  shows  that  the  amplitude  of  a  vibration  is  found  by 
putting  : 


Putting  these  values  in  Eq.  (10)  and  taking  the  difference 
of  the  results,  the  time  of  a  single  oscillation  is  found  to  be : 


Eq.  (11)  is  seen  to  be  identical  with  Eq.  (5).  In  this  case 
the  amplitude  is  2um  and  the  body  oscillates  through  its  posi- 
tion of  rest.  Both  oscillations  are  completely  isochronous  for 
the  same  weight  W. 

If  n  is  the  observed  number  of  oscillations  per  second, 
either  Eq.  (5)  or  (i  i)  gives  : 


£  =  -^  . 


=  n2 


•       -      -      (I*) 


from  which  E  may  be  computed,  if  W  is  very  great  compared 
with  the  weight  of  the  bar  or  wire. 


CHAPTER  IV. 

THEORY    OF    FLEXURE. 

Art.  17. — General  Formulae. 

IF  a  prismatic  portion  of  material  is  either  supported  at  both 
ends,  or  fixed  at  one  or  both  ends,  and  subjected  to  the  action 
of  external  forces  whose  directions  are  normal  to,  and  cut,  the 
axis  of  the  prismatic  piece,  that  piece  is  said  to  be  subjected  to 
"  flexure."  If  these  external  forces  have  lines  of  action  which 
are  oblique  to  the  axis  of  the  piece,  it  is  subjected  to  combined 
flexure  and  direct  stress. 

Again,  if  the  piece  of  material  is  acted  upon  by  a  couple 
having  the  same  axis  with  itself,  it  will  be  subjected  to  "  tor- 
sion." 

The  most  general  case  possible  is  that  which  combines  these 
three,  and  some  general  equations  relating  to  it  will  first  be 
established. 

The  co-ordinate  axis  of  X  will  be  taken  to  coincide  with  the 
axis  of  the  prism,  and  it  will  be  assumed  that  all  external  forces 
act  upon  its  ends  only.  Since  no  external  forces  act  upon  its 
lateral  surface,  there  will  be  taken  : 

r,  =  N2  =  N3  =  o ; 

retaining  tne  notation  of  Art.  6.  These  conditions  are  not 
strictly  true  for  the  general  case,  but  the  errors  are,  at  most, 
excessively  small  for  the  cases  of  direct  stress  or  flexure,  or 


Art.  I/.]  GENERAL  FORMULA.  IO7 

for  a  combination  of  the  two.     By  the  use  of  Eqs.  (12),  (21) 
and  (22)  of  Art.  5,  the  conditions  just  given  become  : 

r         (du    ,    dv   ,    dw\        dv 


-  2r   \dx    '    dy        dz  J        dy 
du        dv        dw\        dw 


=  0 (3) 

dy 


Eqs.  (i)  and  (2)  then  give  : 

—  -  —  =  o  (4) 

dy        dz 

In  consequence  of  Eq.  (4),  Eqs.  (i)  and  (2)  give  : 

dv  __  dw  du  /j.\ 

dy  ~  dz  dx 

By  the  aid  of  Eq.  (5)  and  the  use  of  Eqs.  (11),  (13)  and  (20) 
of  Art.  5,  in  Eqs.  (10),  (u)  and  (12)  of  Art.  6  (in  this  case 
X0  =  F0  =  Z0  =  o),  there  will  result  : 


2  dx>  +  df        dz*  . 

_^_+^  =  o 
dx  dy  T  dx* 


108  THEORY  OF  FLEXURE.  [Art.  I/. 


dx  dz         dx* 


(3) 


The  Eqs.  (3),  (5),  (6),  (7)  and  (8)  are  five  equations  of  con- 
dition by  which  the  strains  u,  v  and  w  are  to  be  determined. 
Let  Eq.  (6)  be  differentiated  in  respect  to  x  : 


—      4_  4. 

2  djfi  ~~  dy*  dx  "  dz*  dx  ~~ 


From  this  equation  let  there  be  subtracted  the  sum  of  the 
results  obtained  by  differentiating  Eq.  (7)  in  respect  toy  and 
(8)  in  respect  to  z  : 


2  __  _    .  _   _    .  _   —   /~\ 

dy        dx2  dz 


In  this  equation  substitute  the  results  obtained  by  differ- 
entiating Eq.  (5)  twice  in  respect  to  x,  there  will  result  : 


This  result,  in  the  equation  immediately  preceding  Eq.  (9) 
by  the  aid  of  Eq.  (5),  will  give  : 


-  —  o. 


.  dx2  dy 

After  differentiating    Eq.  (7)   in   respect  to  y,  and  substi 
tuting  the  value  immediately  above  : 


Art.   17.]  GENERAL  FORMULA. 


ICQ 


Eqs.  (9)  and  (10)  enable  the  second  equation  preceding  Eq. 
(9),  to  give  : 


Let   the  results  obtained  by  differentiating   Eq.  (7)  in  re- 
spect to  z  and  (8)  in  respect  to  y,  be  added  : 


I  ~J.-t       J_       T" 


dx  dy  dz        dx*  dz        dx*  dy 

The  sum  of  the  second  and  third  terms  of  the  first  member 
of  this  equation  is  zero,  as  is  shown  by  twice  differentiating  Eq. 
(3)  in  respect  to  x.  Hence : 


«(£ 


dy  dz  dx          dy  dz 


=  0 (12) 


The  Eqs.  (9),  (10),  (11)  and  (12)  are  sufficient  for  the  de- 
termination of  the  form  of  the  function  —  ,  if  it  be  assumed 
to  be  algebraic,  for  : 

Eq.  (9)  shows  that  x*  does  not  appear  in  it  ; 


"     (12)      "         " 


HO  THEORY  OF  FLEXURE.  [Art.  I/. 

The  products  xz  and  xy  may,  however,  be  found  in  the 
function.  Hence  if  <z,  alt  a2,  b,  b»  and  b2  are  constants,  there 
may  be  written  : 

-r  —  a  +  #i~  ~f*  a*y  ~\~  x  (p  ~\~  b*z  ~\~  b2y)    .     .     (13) 
Eq.  (5)  then,  gives  : 

-r  —  —j-  =  —  r  \a  +  ^i^  +  &*y  +  x  (b  -\-  b^z  +  ^2^)1  (14) 
Substituting  from  Eq.  (13)  in  Eqs.  (7)  and  (8)  : 

£  =-^,-M (I5) 


The  method  of  treatment  of  the  various  partial  derivatives 
in  the  search  for  Eqs.  (13)  and  (14)  is  identical  with  that  given 
by  Clebsch  in  his  "  Theorie  der  Elasticitdt  Fester  Korper" 

It  is  to  be  noticed  that  the  preceding  treatment  has  been 
entirely  independent  of  the  form  of  cross  section  or  direction  of 
external  forces. 

It  is  evident  from  Eqs.  (13)  and  (14),  that  the  constant  a 
depends  upon  that  component  of  the  external  force  which  acts 
parallel  to  the  axis  of  the  piece  and  produces  tension  or  com- 
pression only.  For,  by  Arts.  2  and  3,  it  is  known  that  if  a 
piece  of  material  be  subjected  to  direct  stress  only : 

du  dv       dw 

—  =  a     and     —-  =  —-=  —  ra ; 

dx  dy        dz 


Art.   I/.]  GENERAL  FORMULAE.  Ill 

the  negative  sign  showing  that  ra  is  opposite  in  kind  to  a,  both 
being  constant. 

Again,  if  z  and  y  are   each  equal  to  zero,  Eq.  (13)  shows 
that  : 

du 


Hence  bx  is  a  part  of  the  rate  of  strain  in  the  direction  of  x 
which  is  uniform  over  the  whole  of  any  normal  section  of  the 
piece  of  material,  and  it  varies  directly  with  x.  But  such  a 
portion  of  the  rate  of  strain  can  only  be  produced  by  external 
force,  acting  parallel  to  the-  axis  of  X,  and  whose  intensity 
varies  directly  as  x.  But,  in  the  present  case  such  a  force  does 
not  exist.  Hence  b  must  equal  zero. 

The  Eqs.  (13),  (14),  (15)  and  (16),  show  that  #,,  £,  and  av  b3 
are  symmetrical,  so  to  speak,  in  reference  to  the  co-ordinates  z 
and  y,  while  Eqs.  (13)  and  (14)  show  that  the  normal  intensity 
NI  is  dependent  on  those,  and  no  other,  constants  in  pure 
flexure,  in  which  a  =  o.  It  follows,  therefore,  that  those  two 
pairs  of  constants  belong  to  the  two  cases  of  flexure  about  the 
two  axes  of  Z  and  F. 

No  direct  stress  N^  can  exist  in  torsion,  which   is  simply  a 
twisting  or  turning  about  the  axis  of  X. 

Since  the  generality  of  the  deductions  will  be  in  no  manner 
affected,  pure  flexure  about  the  axis  of  Y  will  be  considered. 
For  this  case  : 

a  —  <?2  =  b~  —  o  =  b. 
Making  these  changes  in  (13)  and  (14)  : 

%  =  aj  +  bs*     .     .     .     /   •     •     (17) 
dx 


112  THEORY  OF  FLEXURE.  [Art.  I/. 


e  =  — 

Also: 


I  — 


.-.     ^  =  2£(r  +  I)  (a,  +  ^>r  -  E(a,  +  ^>       .     (20) 
since  : 

2£(r+  1)  =  ^. 

Taking  the  first  derivative  of  Nl  : 

=  £(a,  +  6s).     .....     (21) 


This  important  equation  gives  the  law  of  variation  of  the 
intensity  of  stress  acting  parallel  to  the  axis  of  a  bent  beam,  in 
the  case  of  pure  flexure  produced  by  forces  exerted  at  its  ex- 
tremity. That  equation  proves,  that  in  a  given  normal  section 
of  the  beam,  whatever  may  be  the  form  of  the  section,  the  rate 
of  variation  of  the  normal  intensity  of  stress  is  constant  ;  the  rate 
being  taken  along  the  direction  of  the  external  forces. 

It  follows  from  this,  that  NT  must  vaiy  directly  as  the  dis- 
tance from  some  particular  line  in  the  normal  section  consid- 
ered in  which  its  value  is  zero.  Since  the  external  forces  F  are 
normal  to  the  axis  of  the  beam  and  direction  of  Niy  and  be- 
cause it  is  necessary  for  equilibrium  that  the  sum  of  all  the 
forces  NI  dy  dz,  for  a  given  section,  must  be  equal  to  zero,  it 


Art.  17.] 


GENERAL  FORMULA. 


follows  that  on  one  side  of  this  line  tension  must  exist  and  on 
the  other,  compression. 

Let  N  represent  the  normal  intensity  of  stress  at  the  dis- 


V 


Fig.1 


tance  unity  from  the  line,  b  the  variable  width  of  the  section 
parallel  toj,  and  let  A  =  b  dz.  The  sum  of  all  the  tensile  stress 
in  the  section  will  be  : 


The  total  compressive  stress  will  be  : 


N 


The  integrals  are  taken  between  the  limits  o  and  the  greatest 
value  of  z  in  each  direction,  so  as  to  extend  over  the  entire 
section.  In  order  that  equilibrium  may  exist  therefore  : 


--0 


(22) 


114  THEORY  OF  FLEXURE.  [Art.  I/. 

Eq.  (22)  shows  that  the  line  of  no  stress  must  pass  througJi  the 
centre  of  gravity  of  the  normal  section. 

This  line  of  no  stress  is  called  the  neutral  axis  of  the  section. 
Regarding  the  whole  beam,  there  will  be  a  surface  which  will 
contain  all  the  neutral  axes  of  the  different  sections,  and  it  is 
called  the  neutral  surface  of  the  bent  beam.  The  neutral  axis 
of  any  section,  therefore,  is  the  line  of  intersection  of  the  plane 
of  section-  and  neutral  surface. 

Hereafter  the  axis  of  X  will  be  so  taken  as  to  traverse  the 
centres  of  gravity  of  the  different  normal  sections  before  flex- 
ure. The  origin  of  co-ordinates  will  then  be  taken  at  the 
centre  of  gravity  of  the  fixed  end  of  the  beam,  as  shown  in. 
Fig.  I. 

The  value  of  the  expression  (#,  -|-  <MO>  m  terms  of  the  ex- 
ternal bending  moment,   is  yet   to  be   determined.     Consider 
any  normal  section  of  the  beam  located  at  the  distance  x  from 
O,  Fig.  i,  and  let  OA  =  I.     Also  let  Fig.  2  rep- 
resent the  section  considered,  in  which  BC  is 
the  neutral  axis  and  d'  and  d^  the  distances  of 
the  most  remote  fibres  from  BC.    Let  moments 
of  all  the  forces  acting  upon  the  portion  (/—  x) 
of  the  beam  be  taken  about  the  neutral  axis 
BC.     If,  again,  b  is  the  variable  width  of  the 
Fig.2  beam,  the  internal  resisting  moment  will  be  : 


fd'  rd> 

NI  bz  dz  =E  (a,  +  b^x)  I 


b  dz. 


But  the  integral  expression  in  this  equation  is  the  moment 
of  inertia  of  the  normal  section  about  the  neutral  axis,  which 
will  hereafter  be  represented  by  /.  The  moment  of  the 
external  force,  or  forces,  Fy  will  be  F(l—x]  and  it  will  be 
equal,  but  opposite  in  sign,  to  the  internal  resisting  moment. 
Hence  : 


Art'  !7-]  GENERAL  FORMULA.  1 15 

.      .     .     (23) 


Substituting  this  quantity  in  Eq. 


UNIVERSITY 

V 


It  will  hereafter  be  seen  that  Eq.  (25)  is  one  of  the  most 
important  equations  in  the  whole  subject  of  the  "Resistance  of 
Materials:' 

An  equation  exactly  similar  to  (25)  may,  of  course,  be 
written  from  Eq.  (16)  ;  but  in  such  an  expression  M  will  repre- 
sent the  external  bending  moment  about  an  axis  parallel  to 
the  axis  of  Z. 

No  attempt  has  hitherto  been  made  to  determine  the  com- 
plete values  of  u,  v,  and  w,  for  the  mathematical  operations 
involved  are  very  extended.  If,  however,  a  beam  be  considered 
whose  width,  parallel  to  the  axis  of  F,  is  indefinitely  small  u 
and  w  may  be  determined  without  difficulty.  The  conclusions 
reached  in  this  manner  will  be  applicable  to  any  long  rectan- 
gular beam  without  essential  error. 

If  y  is  indefinitely  small  all  terms  involving  it  as  a  factor 
will  disappear  in  u  and  w  ;  or,  the  expressions  for  the  strains  u 
and  w  will  be  functions  of  z  and  x  only.  But  making  u  and  w 
functions  of  z  and  x  only  is  equivalent  to  a  restriction  of  lateral 
strains  to  the  direction  of  z  only,  or,  to  the  reduction  of  the 
direct  strains  one  half,  since  direct  strains  and  lateral  strains  in 
two  directions  accompany  each  other  in  the  unrestricted  case. 
Now  as  the  lateral  strain  in  one  direction  is  supposed  to  retain 
the  same  amount  as  before,  while  the  direct  strain  is  considered 
only  half  as  great,  the  value  of  their  ratio  for  the  present  case 


Il6  THEORY  OF  FLEXURE.  [Art.  I/. 

will  be  twice  as  great  as  that  used  in  Arts.  3  and  4.  Hence  2r 
must  be  written  for  r,  in  order  that  that  letter  may  represent 
the  ratio  for  the  unrestricted  case,  and  this  will  be  done  in  the 
following  equations. 

Since  w  and  u  are  independent  of  y  : 

dw       du  ~dv 

—   ""     ~  • 


-J-        -r  >  3  — 

dy        dy  dx 

But  by  Eq.  (14)  : 

v  =  -  2r  (a,  +  £f*)* 
By  Eq.  (3),  since  : 


dw 

—  o  • 

- 


=  -  2r.  (a, 


This  equation,  however,  involves  a  contradiction,  for  it 
makes  f(xj  z)  equal  to  a  function  which  involves/*  which  is 
impossible.  Hence  : 

/(*,  e)  =  o. 
Consequently  : 


which  is  indefinitely  small  compared  with  : 

dv  / 

_  =  -  27-  (aT 

and  is  to-be  considered  zero. 


Art.  17.]  GENERAL  FORMULAE. 

Because /(.r,  z)  =  o  : 


dv 
- 


This  quantity  is  indefinitely  small  ;  hence  : 
T3  =  -  zGrbjsy 

is  of  the  same  magnitude. 

Under  the  assumption  made  in  reference  to  y,  there  may 
be  written  from  Eqs.  (17)  and  (18)  : 


(26) 


w  =  -  ra  ^cs  ....     (27) 

Using  Eq.  (26)  in  connection  with  Eq.  (6)  : 


By  two  integrations  : 

/'(,)=-*£  -c's  +  c" (28) 

Using  Eq.  (27)  in  connection  with-Eq.  (8)  : 


By  two  integrations  : 


Il8  THEORY  OF  FLEXURE.  [Art. 


ft-\  7     X  *  • 

/to    ^    "    *i  --    -     --  +   ^  +  *i«- 


The  functions  #  and  w  now  become  : 


u  —  a^xz  -j-  £x  —  ^  —  —  --  ^  +  c"  .     .     .     (29) 


csl  .     (30) 


The  constants  of  integration  c ',  c",  etc.,  depend  upon  the 
values  of  u  and  w,  and  their  derivatives,  for  certain  reference 
values  of  the  co-ordinates  x  and  2,  and,  also,  upon  the  manner 
of  application  of  the  external  forces,  F,  at  the  end  of  the  beam, 
Fig.  I.  The  last  condition  is  involved  in  the  application  of 
Eqs.  (13),  (14)  and  (15)  of  Art.  6. 

In  Fig.  i  let  the  beam  be  fixed  at  O.  There  will  then  re- 
sult for  x  .—  o  and  z  =  o : 


(du 
\dz 


=  0} 

JX=0 

'     Z  =0 

(u  —  o,     and     w  —  o) 


In  virtue  of  the  last  condition  : 

c"  =  c,,  =  o. 
In  consequence  of  the  first  : 

c'  =o. 
After  inserting  these  values  in  Eqs.  (29)  and  (30) 


Art.  I/.]  GENERAL  FORMULAE.  1  19 

du  x*  • 


=  -  rbj?  -  b,        -  a,x  +  c,  ; 


The  surface  of  the  end  of  the  beam,  on  which  F  is  applied, 
is  at  the  distance  /  from  the  origin  O  and  parallel  to  the  plane 
ZF.  Also  the  force  F  has  a  direction  parallel  to  the  axis  of 
Z.  Using  the  notation  of  Eqs.  (13),  (14)  and  (15)  of  Art.  6, 
these  conditions  give  : 

cos  p  —  I,         cos  q  =  O,         cos  r  =  o, 
cos  7t  —  O,         cos  x  =  O,         cos  p  =  I. 
Since  for  x  —  /  : 

M  =  F(l  -  x)  =  o, 

Eqs.  (24)  and  (20)  give  NT  =  o  for  all   points  of  the  end  sur- 
face.    Eq.  (15)  is,  then,  the  only  one  of  those  equations  which 
is  available  for  the  determination  of  cr 
That  equation  becomes  simply  : 

7;  =  P. 

For  a  given  value  of  z,  therefore,  any  value  maybe  assumed 
for  T2.     For  the  upper  and  lower  surfaces  of  the  beam  let  the 
intensity  of  shear  be  zero  ;  or  for  z  -  ±  d  let  T2  =  o. 
byEq.(3i): 


120  THEORY  OF  FLEXURE.  [Art.  17. 


(32) 


The  constants  «x  and  £x  still  remain  to  be  found.  The  only 
forces  acting  upon  the  portion  (/  —  x)  of  the  beam,  are  F  and 
the  sum  of  all  the  shears  Ta  which  act  in  the  section  x.  Let 
Aj  be  the  indefinitely  small  width  of  the  beam,  which,  since 
z  is  finite,  is  thus  really  made  constant.  The  principles  of 
equilibrium  require  that  : 

(d*  .  ky.dz  —  s>  .  &y.dz)*=F. 


The  first  part  of  the  integral  will  be  2  Ajj/^3  and  the  second 
part  will  be  the  moment  of  inertia  of  the  cross  section  (made 
rectangular  by  taking  A/  constant)  about  the  neutral  axis. 
Hence  : 


i+r)f--      -F;    or    I,  =  2G(l  +  ry  -- ~-  ^    (33) 

••'    ^=£(^-^) (34) 

If  x  =  o  in  Eq.  (24)  : 


Thus  the  two  conditions  of  equilibrium  are  involved  in  the 
determination  of  <&,  and  bv  The  complete  values  of  the  strains 
u  and  w  are,  finally  : 


Art.  17.]  GENERAL  FORMULA.  121 


Fd*x 
w  =  ^.\ire^r*r-^±~~Y+-j^.    .    (37) 

These  results  are  strictly  true  for  rectangular  beams  of  in- 
definitely small  width,  but  they  may  be  applied  to  any  rectan- 
gular beam  fixed  at  one  end  and  loaded  at  the  other,  with 
sufficient  accuracy  for  the  ordinary  'purposes  of  the  civil  en- 
gineer. It  is  to  be  remembered  that  the  load  at  the  end  is 
supposed  to  be  applied  according  to  the  law  given  by  Eq.  (34) ; 
a  condition  which  is  never  realized.  Hence  these  formulae  are 
better  applicable  to  long  than  short  beams. 

The  greatest  value  of  Tv  in  Eq.  (34),  is  found  at  the  neutral 
axis  by  making  z  =  o ;  for  which  it  becomes  : 

.  3      F 


T  - 
a"  2 


—,  is  the  mean  intensity  of  shear  in  the  cross  section  ;  hence, 

the  greatest  intensity  of  shear  is  once  and  a  half  as  great  as  the 
mean. 

In  Eq.  (36)  if  z  =  o,  u  =  o.  Hence  no  point  of  the  neutral 
surface  suffers  longitudinal  displacement. 

In  Eq.  (37)  the  last  term  of  the  second  member  is  that  part 
of  the  vertical  deflection  due  to  the  shear  at  the  neutral  sur- 
face, as  is  shown  by  Eq.  (38).  The  first  term  of  the  second 
member,  being  independent  of  x,  is  that  part  of  the  deflection 
which  arises  wholly  from  the  deformation  of  the  normal  cross 
section. 

The  usual  modification  of  the  preceding  treatment,  designed 
to  supply  formulae  for  the  ordinary  experience  of  the  engineer, 
will  be  given  in  the  succeeding  Arts. 


122  THEORY  OF  FLEXURE.  [Art.  1 8. 


Art.  18.— The  Common  Theory  of  Flexure. 

The  "  common  theory  "  of  flexure  is  completely  expressed 
by  Eq.  (25)  of  Art.  17.  That  equation  involves  the  condition 
that  no  external  force  acts  ^lpon  the  exterior  surface  of  tJie  bar  or 
beam.  In  reality  this  condition  is  never  fulfilled.  External 
loads  are  applied  in  any  manner  whatever,  causing  normal 
compressive  stresses  to  exist  at  any  or  all  points  of  the  ex- 
terior surface.  //  is  assumed  in  the  common  theory  of  flexure 
that  the  equation  : 

**w        M 


holds  true,  for  pure  bending,  whatever  may  be  the  number  or 
manner  of  application  of  the  external  forces  or  loads. 

By  "  pure  bending  "  is  meant  the  action  of  external  forces 
whose  directions  are  normal  to,  and  cut,  the  axis  of  the  beam. 

As  has  already  been  seen  in  Art.  17,  w,  strictly  speaking,  is 
a  function  of  x,  y  and  z. 

It  is  further  assumed  in  the  common  theory  of  flexure  that  w 
is  a  function  of  x  only. 

This  is  equivalent  to  an  assumption  that  the  lateral  dimen- 
sions of  the  piece  are  so  small  that  they  can  have  no  influence 
on  the  value  of  w,  and  consequently  that  they  will  not  appear 
in  it.  In  other  words  the  common  theory  of  flexure  is  the 
theory  of  the  flexure  of  pieces,  one  or  two  of  whose  cross 
dimensions  are  indefinitely  small  in  comparison  with  their  length. 
The  neglect  of  this  fact  has  led  to  some  erroneous  applications 
and  deductions  in  connection  with  long  column  formulae. 

Eq.  (i),  taken  in  connection  with  these  two  important  as- 
sumptions, constitutes  the  "  Common  Theory  of  Flexure/* 
which  is  always  used  in  engineering  practice. 


Art.   1  8.]          THE  COMMON  THEORY  OF  FLEXURE.  12$ 

Since  the  intensity  of  external  loading  is  almost  invariably 
very  small  compared  with  the  internal  stress  Nlt  the  first  of  the 
above  assumptions  involves  very  little  error  in  all  ordinary 
cases. 

The  second  assumption,  as  was  stated  above,  is  equivalent 
to  taking  the  bar  or  beam  so  small  that  the  strain  or  "  deflec- 
tion "  w  is  essentially  the  same  at  all  points  of  a  given  cross 
section.  With  such  small  strains  and  large  ratios  of  length  to 
lateral  dimensions  as  almost  always  occur,  this  assumption, 
also,  involves  no  considerable  error. 

It  is  well  known  that  if  the  curvature  is  very  small,  the  re- 
ciprocal of  the  radius  of  curvature,  in  the  plane  zx,  is  repre- 

sented with  no  essential   error  by  -=--.     Hence  Eq.  (i)  may 
take  the  form  : 


(2) 


in  which  p  is  the  radius  of  curvature. 

Let  M'  and  M,  represent  two  bending  moments  which  will 
produce  the  two  radii  of  curvature  p'  and  p,.  Eq.  (2)  will  then 
give  the  following  : 


=&•    ........    (3) 


£i  =  j|fs (4) 

A 

Hence  : 

,  v 

—     J.VJ.  T      """^      ^rx  •  •  •  •  '  \3/ 


124 


THEORY  OF  FLEXURE. 


[Art.  1 8. 


The  second  member  shows  that  a  bending  moment : 

MI  -  M'  =  M, 

applied  to  a  curved  beam  whose  radius  of  curvature  at  any 
section  is  p',  will  produce  a  change  of  curvature  expressed  by  : 


In  other  words :  the  common  theory  of  flexure  is  applicable  to 
curved  beams  of  slight  curvature. 

In  such  a  case  — ,  Eq.  (2),  expresses  the  variation  (increase 

or  decrease)  of  curvature  caused  by  the  moment  M.  It  is  to 
be  distinctly  borne  in  mind,  however,  that  Eq.  (2)  itself  is 
made  approximately  true  only  by  considering  the  curvature 
very  small. 

The  limits  within  which  the  common  theory  is  applicable  to 

curved  beams,  and  the  degree 
of  approximation  of  the  appli- 
cation, will  be  shown  by  the 
following  investigations,  in 
which  the  longitudinal  com- 
pression and  extension,  due  to 
the  external  forces,  will  be 
neglected. 

In  the  figure  let  a  portion 
of  any  curved  beam,  whose 
lateral  dimensions  are  small 
compared  with  its  length,  be 


Fig.1 


represented.  Let  AB  represent  an  indefinitely  short  length, 
ds,  of  the  neutral  surface.  C  is  the  centre  of  curvature  of  ds 
before  flexure,  and  C'  the  same  point  after  flexure.  Since  the 


Art  1 8.]  THE  COMMON  THEORY  OF  FLEXURE.  12$ 

lateral  dimensions  are  small  compared  with  the  length,  if  the 
strains  are  not  great,  any  normal  cross  section  may,  without 
essential  error,  be  taken  as  plane  after  flexure,  and  such  planes 
passing  through  A  and  B  will  then  contain  the  radii  of  curva- 
ture at  the  points  A  and  B.  Let  : 

AC'  =  p'    and    AC  =  p 
also  : 

Aa  —  Ab  =  Be  —  Bd  =    unity. 

Aa  and  Bd  are  the  positions  taken  by  Ab  and  Be  after  flexure. 
The  angle,  before  flexure,  between  two  radii  AC  and  BC,  in- 
definitely near  to  each  other,  is  — ;  after  flexure,  as  the  figure 

shows,   the   same  angle   becomes  —  .     Hence  the  change   in 

curvature  (or  change  of  angle  between  consecutive  radii) 
caused  by  flexure  is  : 


Now  let  the  amount  of  shortening  or  lengthening  of  a  unit 
of  length  of  fibres,  parallel  to  the  neutral  surface  and  situated 
at  unit's  distance  from  it,  be  represented  by  u]  concisely  stated, 
u  is  the  rate  of  strain  for  any  point  at  unit's  distance  from  the 
neutral  surface.  In  the  figure,  the  amount  of  strain  for 
AB  =  ds  is  : 

ab  -f-  cd  =  u  ds. 
But  the  difference  between  the  angles  aC'd  and  bCc  is  : 

(ab  +  cd)  -*•  Ab  =  ab  +  cd  =  u  ds. 


126  THEORY  OF  FLEXURE.  [Art.  1 8. 

But  this  difference  is  the  change  of  curvature  ;  hence  : 


This  relation  is  purely  kinematical ;  a  value  for  u  must  next 
be  determined  in  terms  of  the  bending  moment  M. 

Under  the  circumstances  of  the  case  it  has  been  seen  that 
the  longitudinal  strains  parallel  to  the  neutral  surface  vary  es- 
sentially directly  as  their  distances  from  it  (this  law  is  the  as- 
sumption that  plane  normal  sections  before  flexure  are  also 
plane  afterwards).  The  strain  at  any  distance  z  from  the 
neutral  surface  will  then  be  uz.  But  it  was  shown  in  Art.  17 
that  the  intensity  of  longitudinal  stress  ^V,  varies  directly  as  z ; 
hence  there  may  be  written  : 

NI  —  Euz. 

If  b  is  the  variable  width  of  cross  section,  taken  parallel  to 
the  neutral  surface,  the  internal  resisting  moment  of  the  sec- 
tion will  be  : 

M  =   \Ni  b  dz  .  z  —  Eu  bz2  dz. 
•••    M=EuI ,...    (7) 


The  integration  is  to  be  extended  over  the  whole  section. 
Then,  if  the  "  neutral  axis  "  is  the  line  of  intersection  of  the 
neutral  surface  with  the  normal  section,  I  is  the  moment  of 
inertia  of  the  normal  section  about  the  neutral  axis. 


Art.  1  8.]  THE  COMMON  THEORY  OF  FLEXURE.  12J 

Eqs.  (6)  and  (8)  then  give  : 

M         i          I 
~EI~  7  '     ~P 

This  equation  is  true,  under  the  assumption  made,  for  any 
degree  of  curvature  whatever  in  the  original  beam. 

If  w  and  x  are  rectangular  co-ordinates  in  the  plane  of  the 
beam,  x  being  the  independent  variable,  the  expressions  for 
the  reciprocals  of  the  radius  of  curvature  before  and  after 
flexure,  are  : 


,   . 
7  :  :  ~d          ' 


By  the  binomial  formula  : 


and  an  exactly  similar  expression  for  —,.    After  introducing 

these  in  Eqs.  (10)  and  (11),  and  supposing  the  deflections  to  be 
small,  there  may  be  written  : 


_  i         _ 

'     ~        '  ~ 


?        dw'*\  d*w' 
' 


_ 
2   \dx*         dx*      dx* 


15  _  , 

"  dx* 


128  THEOR  Y  OF  FLEXURE.  [Art.  1  8. 

If,  in  addition  to  small  deflections,  the  values  of  : 

dw'  ,       dwi 

-j—,      and       —  JT-, 
dx  dx 

are  not  great,  the  equation  just  written  shows  that  with  a  con- 
siderable degree  of  approximation  : 


p'         p   '      dx*          dx2 


The  smaller  the  curvature  the  more  nearly  accurate  is  Eq. 
(12).     If,  as  before,  w  is  the  deflection  or  strain  normal  to  x  : 


hence,  from  Eqs.  (9)  and  (12)  : 

M 


El       dx* 


(13) 


Eq.  (13)  is  exactly  the  same  as  Eq.  (i)  for  straight  beams. 

These  investigations  show  that  the  common  theory  of  flex- 
ure is  not  strictly  applicable  to  the  general  case  of  curved 
beams.  In  order  to  obtain  Eq.  (12)  it  was  necessary  to  assume 
the  same  law  for  stresses  and  strains,  in  any  normal  section, 
both  for  curved  and  straight  beams,  which  is  not  exactly  true. 
It  was  also  necessary  to  assume  small  values  of 

dWi  ,  dw' 
-~  and  -=— 
dx  dx 


Art.  1  8.]          THE  COMMON  THEORY  OF  FLEXURE.  1  29 

for  a  close  approximation.  Yet  the  application  of  the  common 
theory  of  flexure  to  curved  beams,  even  within  these  restricted 
limits,  is  of  the  highest  importance. 

In  Art.  22  a  generalization  of  the  common  theory  of  flexure 
is  given,  in  which  the  differential  of  the  centre  line  of  the  beam 
is  used  instead  of  dx.  The  resulting  formulae  are  accurately 
applicable  to  curved  beams  of  any  curvature.  The  only  as- 
sumption involved,  in  addition  to  those  of  the  common  theory, 
is  the  identity  of  the  law  of  variation  of  stresses  and  strains  in 
curved  and  straight  beams  ;  and  that  causes  very  little  error. 

One  of  the  most  important  forms  of  Eq.  (7)  yet  remains  to 
be  established. 

Let  d^  represent  the  distance  from  the  neutral  axis  of  any 
normal  section  of  the  beam  to  that  point  of  the  section  farthest 
from  it.  Let  K  represent  the  intensity  of  tensile  or  compres- 
sive  stress  (as  the  case  may  be)  existing  at  this  same  point  ;  K 
will  be  the  greatest  intensity  in  the  section.  Because  the  in- 
tensity of  stress  varies  directly  as  the  distance  from  the  neutral 
axis,  the  intensity  at  distance  unity  from  that  axis  will  be  : 


But  by  Art.  2,  this  intensity  also  has  the  value  Eu.     Con- 
sequently Eq.  (7)  becomes  : 


(14) 


If  the  external  moment  is  sufficient  to  break  the  beam,  and 
if  Eq.  (14)  is  applied  to  the  section  at  which  failure  begins,  K 
is  called  the  "  Modulus  of  Rupture"  for  flexure.  It  is  an 
empirical  quantity. 


13°  THEORY  OF  FLEXURE.  [Art.  19. 


Art.  19.— Deflection  by  the  Common  Theory  of  Flexure. 

The  common  theory  of  flexure,  as  developed  in  the  pre. 
ceding  Art.,  leads  to  very  simple  and,  in  nearly  all  ordinary 
cases,  very  closely  approximate  formulae. 

Let  x0  be  the  co-ordinate  of  some  point  at  which  the  tan- 
gent of  the  inclination  of  the  neutral  surface  to  the  axis  of  x 
is  known  ;  then,  from  Eq.  (i)  of  Art.  18  : 

dw 
'dx  ~ 


/JfJQJ 

——  will  be  at  once  recognized  as  the  general  value  of  the  tan- 
gent of  the  inclination  just  mentioned,  or,  in  the  case  of  curved 
beams,  as  approximately  the  difference  between  the  tangent, 
before  and  after  flexure. 

Again,  let  x^  represent  the  co-ordinate  of  a  point  at  which 
the  deflection  w  is  known,  then,  from  Eq.  (i) : 


9-r^ (2) 


The  points  of  greatest  or  least  deflection  and  greatest  or 
least  inclination  of  neutral  surface  are  easily  found  by  the  aid 
of  Eqs.  (i)  and  (2). 

The  point  of  greatest  or  least  deflection  is  evidently  found 
by  putting  : 

dw 


dx 


(3) 


and  solving  for  x.     Since  — - —  is  the  value  of  the  tangent  of  the 


OF   THE          'V   \ 

^UNIVERSITT^ 

Art  *9-]  DEFLECTION. 


inclination  of  the  neutral  surface,  it  follows  that  a  point  of 
greatest  or  least  deflection  is  found  where  the  beam  is  hori- 
zontal. 

Again,  the  point  at  which  the  inclination  will  be  greatest  or 
least  is  found  by  the  equation  : 


dzw 


//    7fJ 

But,  approximately,  -—  is  the  reciprocal  of  the  radius  of 

curvature  ;  hence,  the  greatest  inclination  will  be  found  at  that 
point  at  which  the  radius  of  curvature  becomes  infinitely  great, 
or,  at  that  point  at  which  the  curvature  changes  from  positive 
to  negative  or  vice  versa.  These  points  are  called  points  of 
"  contra-flexure."  Since  : 

M-  El  -- 

dx*  ' 


there  is  no  bending  at  a  point  of  contra-flexure. 

The  moment  of  the  external  forces,  M,  will  always  be  ex- 
pressed in  terms  of  x.  After  the  insertion  of  such  values,  Eqs. 
(i)  and  (2)  may  at  once  be  integrated  and  (3)  and  (4)  solved. 

The  coefficient  of  elasticity,  E,  is  always  considered  a  con- 
stant quantity  ;  hence  it  may  always  be  taken  outside  the  in- 
tegral signs.  In  all  ordinary  cases,  also,  /  is  constant  through- 
out the  entire  beam.  In  such  cases,  then,  there  will  only  need 
to  be  integrated  the  expressions  : 


[*  Mdx     and      f*   [*  M  d*. 

*  X~  *  X-    J  X 


132  THEORY  OF  FLEXURE.  [Art.  2O. 

Before  applying  these  formulae  to  particular  cases  it  will  be 
necessary  to  consider  some  other  matters. 


Art.  20.  —  External  Bending  Moments  and  Shears  in  General. 

Beams  subjected  to  combined  bending  and  direct  stress  will 
not  be  treated.  Such  cases  are  of  little  or  no  real  value  to  the 
engineer,  and  approximate  solutions,  even,  are  only  to  be 
reached  by  the  higher  processes  of  analysis.  In  all  beams, 
therefore,  pure  bending  only  is  to  be  treated.  A  beam  is  said 
to  be  non-continuous  if  its  extremities  simply  rest  at  each  end 
of  the  span  and  suffer  no  constraint  whatever. 

A  beam  is  said  to  be  continuous  if  its  length  is  equal  to  two 
or  more  spans,  or  if  its  ends,  in  case  of  one  span  (or  more)  suffer 
constraint. 

A  cantilever  is  a  beam  which  overhangs  its  span  ;  one  end  of 
which  is  in  no  manner  supported.  Each  of  the  overhanging 
portions  of  an  open  swing  bridge  is  a  cantilever  truss. 

Let  any  beam  be  horizontal,  and  suppose  it  to  be  subjected 
to  vertical  loads.  The  results  will  evidently  be  applicable  to 
any  beam  acted  upon  by  loads  normal  to  its  axis.  Let  P  be 
any  single  vertical  load,  and  let  x  be  any  horizontal  co-ordinate 
measured  from  any  point  as  an  origin.  Let  &\  represent  the 
co-ordinate,  measured  from  the  same  origin,  of  the  point  of  ap- 
plication of  any  load  P.  Finally,  let  it  be  required  to  deter- 
mine the  external  bending  moment  J/at  any  section,  x,  of  the 
beam.  The  lever  arm  of  any  load  P  is  evidently  (x  —  x^). 

Hence,  for  any  number  of  forces  : 


(i) 


The  summation  sign  2  refers  only  to  x^  and  is  to  cover  that 
portion  of  the  beam  on  one  side  of  the  section  x,  as  is  evident 
from  the  manner  of  forming  the  equation. 


Art.  2O.]  MOMENTS  AND   SHEARS. 


133 


If  the  origin  of  x  is  in  the  section  considered  : 

M  =.  - 
From  Eq.  (i)  : 

dM 


dx 


(3) 


Now  2P  =  Sis  the  algebraic  sum  of  all  the  forces  on  one 
side  of  the  section  considered,  it  is  consequently  the  total  force 
acting  in  the  section  tending  to  move  one  portion  of  tJie  beam  past 
the  other;  it  is  therefore  called  the  "  shear  "  in  the  section. 
This  quantity  (the  shear)  is  a  most  important  one  in  the  sub- 
ject of  the  resistance  of  materials. 

The  reactions,  or  supporting  forces,  applied  to  the  beam, 
are  to  be  included  both  in  the  sum  2P,  and  in  the  moment  : 


Eq.  (3)  shows  that  the  shear  at  any  section  is  equal  to  the  first 
differential  coefficient  of  the  bending  moment  considered  as  a 
function  of  x. 

The  sum  of  all  the  loads  on  the  other  side  of  the  section  x 
would  give  the  same  numerical  shear,  but  it  would  evidently 
have  an  opposite  direction. 

As  is  well  known,  the  analytical  condition  for  a  maximum 
or  minimum  bending  moment  in  a  beam  is  : 


o    ........    (4) 

dx 


From   Eqs.  (3)  and  (4)   is  to  be  deduced  the  following  im- 


134  THEORY  OF  FLEXURE.  [Art.  2O. 

portant  principle  :  The  greatest  or  least  bending  moment  in  any 
beam  is  to  be  found  in  that  section  for  which  the  shear  is  zero. 

The  importance  of  this  principle  lies  in  the  fact  that  in  the 
greater  portion  of  cases  of  loaded  beams  which  come  within 
the  experience  of  the  civil  engineer,  the  section  subjected  to 
the  greatest  bending  moment  can  thus  be  determined  by  a 
simple  inspection  of  the  loading. 

These  principles  can  be  well  illustrated  by  the  following 
simple  example. 

Fig.  i    represents  a  non-continuous  beam  with  the  span  /, 


5= 


FIG.  i. 

supporting  two  equal  weights  P,  P.  These  two  weights  or 
loads  are  to  be  kept  at  a  constant  distance  apart  denoted 
by  a. 

It  is  required  to  find  that  position  of  the  two  loads  which 
will  cause  the  greatest  bending  moment  to  exist  in  the  beam, 
and  the  value  of  that  moment.  The  reaction  R  is  to  be  found 
by  the  simple  principle  of  the  lever.  Its  value  will  therefore 
be: 


(5) 


Since  the  reaction  R  can  never  be  equal  to  2.P,  2P,  or  the 
shear,  it  must  be  equal  to  zero  at  the  point  of  application  of  one 
of  the  loads  P.  In  searching  for  the  greatest  moment,  then, 
it  will  only  be  necessary  to  find  the  moment  about  the  point 


Art.  2O.]  MOMENTS  AND   SHEARS.  135 

of  application  of  one  of  the  forces  P.     It  will  be  most  con- 
venient to  take  that  one  nearest  R. 
The  moment  desired  will  be  : 


*--  .    .    .    .    (6) 


2          4 

This  value  in  Eq.  (6)  gives  : 


Since  : 


it  appears  that  M,  is  a  maximum. 

If  the  load  is  uniformly  continuous  and  of  the  intensity/, 

in  Eqs.  (i),  (2)  and  (&pdx^  is  to  be  put  for  P,  and  the  sign  J  for 
2.     Hence  : 


M  = 


=  -  /J  ^i  ^-ri  • 


136  THEORY  OF  FLEXURE.  [Art.  2O. 

dM 


But  since  dx  and  dxT  are  perfectly  arbitrary,  they  may  be 
taken  equal  to  each  other,  hence  : 


Or,  the  second  differential  coefficient  of  the  moment,  considered 
as  a  function  of  xy  is  equal  to  the  intensity  of  the  continuous 
load. 

A  very  important  problem  arises  in  connection  with  the 
principles  discussed  in  this  Art.  It  is  the  following : 

A  continuous  train  of  any  given  varying  or  uniform  density 
advances  along  a  simple  beam  of  span  I.  It  is  required  to  deter- 
mine what  position  of  loading  will  give  the  greatest  shear  at  any 
'specified  section. 

In  Fig.  2,  CD  is  the  span  /,  and  A  is  any  section  for  which 
c  A  B  D  ^  is  required  to  find  the 

position  of  the  load  for 


Fi£'2  ™    the  greatest  transverse 

shear.  The  load  is  supposed  to  advance  continuously  from  C 
to  any  point  B.  Let  R  be  the  reaction  at  D,  and  2P  the  load 
between  A  and  B.  The  shear  S'  at  A  will  be  : 


R  -  2P  =  S'    .......     (8) 

Let  R'  be  that  part  of  R  which  is  due  to  2P,  and  R"  that 
part  due  to  the  load  on  CA  ;  evidently  R'  is  less  than  2  P. 
Then  : 


R'  +  R"  -  2P=  S'. 


Art.  20.]  MOMENTS  AND   SHEARS.  137 

If  AB  carries  no  load,  R  and  .21/*  disappear  in  the  value  of 
5.  Hence  : 

R"  =  S 

is  the  shear  for  the  head  of  the  train  at  A.  S  is  greater  than 
S'  because  2P  is  greater  than  R.  But  no  load  can  be  taken 
from  AC  without  decreasing  R".  .  Hence:  The  greatest  shear 
at  any  section  will  exist  when  the  load  extends  from  the  end  of 
the  span  to  that  section,  whatever  be  the  density  of  the  load. 

In  general,  the  section  will  divide  the  span  into  two  un- 
equal segments.  The  load  also  may  approach  from  either 
direction.  The  greater  or  smaller  segment,  then,  may  be 
covered,  and,  according  to  the  principle  just  established,  either 
one  of  these  conditions  will  give  a  maximum  shear.  A  con- 
sideration of  these  conditions  of  loading  in  connection  with 
Fig.  2,  however,  wilt  show  that  these  greatest  shears  will  act  in 
opposite  directions. 

When  the  load  covers  the  greater  segment  the  shear  is 
called  a  main  shear;  when  it  covers  the  smaller,  it  is  called  a 
counter  shear. 

Again,  let  a  continuous  load  of  any  given  varying  or  con- 
stant density  traverse  a  non-continuous  beam  ;  it  is  required  to 
find  what  position  of  the  load  will  cause  the  greatest  bending 
moment  at  any  specified  section. 

Since  every  load  which  can  come  upon  the  beam  will  pro- 
duce the  same  kind  of  bending  at  any  given  section,  the  greatest 
possible  amount  of  load  must  be  brought  on  the  beam  for  the 
greatest  bending  moment  at  any  section. 

Hence,  the  greatest  bending  moment,  at  the  specified  section, 
will  exist  when  the  load  covers  the  whole  span. 

It  also  follows  that  all  sections  will  suffer  their  greatest 
bending  moments  with  the  same  position  of  load. 

The  principles  involved  in  these  results  find  important  ap- 
plications in  the  theory  of  truss  bridges. 


138  THEORY  OF  FLEXURE.  [Art.  21. 


Art.  2i.—  Moments  and  Shears  in  Special  Cases. 

Certain  special  cases  of  beams  are  of  such  common  occur- 
rence, and  consequently  of  such  importance,  that  a  somewhat 
more  detailed  treatment  than  that  already  given  may  be 
deemed  desirable.  The  following  cases  are  of  this  character. 

Case  I. 

Let  a  non-continuous  beam,  supporting  a  single  weight  P 

at  any  point,  be  considered, 
and  let  such  a  beam  be  rep- 
resented   in    Fig.    i.     If   the 
'      span  RR  is  represented  by 


I  =:a  +  b  =  RP+  RP, 
the  reactions  R  and  R'  will  be  : 


R  =  -  P,  and    R  =  ~  P (i) 


Consequently,  if  x  represents  the  distance  of  any  section  in 
RP  from  R,  while  x  represents  the  distance  of  any  section  of 
R'P  from  R',  the  general  values  of  the  bending  moments  for 
the  two  segments  a  and  b  of  the  beam  will  be : 

•M  =  Rx,  and  M'  =  Rx' (2) 

These  two  moments  become  equal  to  each  other  and  repre- 
sent the  greatest  bending  moment  in  the  beam  when 

x  =  a  and  X*  =  b 


Art.  21.]  SPECIAL  MOMENTS  AND   SHEARS. 


139 


or,  when  the  section  is  taken  at  the  point  of  application  of  the 
load  P. 

Eq.  (2)  shows  that  the  moments  vary  directly  as  the  dis- 
tances from  the  ends  of  the  beam.  Hence,  if  AP  (normal  to 
RR')  is  taken  by  any  convenient  scale  to  represent  the  greatest 

moment,  —  P,  and  if  RAR'  is  drawn,  any  intercept  parallel  to 

AP  and  lying  between  RAR'  and  RR  will  represent  the  bend- 
ing moment  for  the  section  at  its  foot,  by  the  same  scale.  In 
this  manner  CD  is  the  bending  moment  at  D. 

The  shear  is  uniform  for  each  single  segment ;  it  is  evi- 
dently equal  to  R  for  RP  and  R'  for  R'P.  It  becomes  zero  at 
P,  where  is  found  the  greatest  bending  moment. 

Case  II. 


Again,  let  Fig.  2  represent  the  same  beam  shown  in  Fig.  I, 
but  let  the  load  be  one  of  uniform  intensity,/,  extending  from 
end  to  end  of  the  beam.  Let  C  be  placed  at  the  centre  of  the 
span,  and  let  R  and  R',  as  be-  A 

fore,  represent  the  two  reac- 
tions. Since  the  load  is  sym- 
metrical in  reference  to  C, 


=  R. 


For  the  same  reason  the  mo- 
ments and  shears  in  one  half  of  the  beam  will  be  exactly  like 
those  in  the  other ;  consequently,  reference  will  be  made  to 
one  half  of  the  beam  only.  Let  x  and  *,  then  be  measured 
from  R  toward  C.  The  forces  acting  upon  the  beam  are  R 
and  /,  the  latter  being  uniformly  continuous.  Applying  the 
formulae  of  the  preceding  Art.,  the  bending  moment  at  any 
section  x  will  be  : 


I4O  THEORY  OF  FLEXURE.  [Art.  21 


»«*r- 


-^ (3) 

If  /  is  the  span,  at  C,  M  becomes  : 

But  because  the  load  is  uniform  : 

~2~  ' 

Hence  : 


if  W  is  put  for  the  total  load.     Placing  : 

in  Eq.  (3)  : 

M=£(lx-  x2}  .  .     (6) 

^        \  /  N         ' 


The  moments  J/,  therefore,  are  proportional  to  the  abscis- 
sae of  a  parabola  whose  vertex  is  over  C,  and  which  passes 
through  the  origin  of  co-ordinates  R.  Let  A  C,  then,  normal 
to  RR',  be  taken  equal  to  Mlt  and  let  the  parabola  RAR  be 


Art.  21.]  SPECIAL   MOMENTS  AND   SHEARS.  141 

drawn.     Intercepts,  as  FH,  parallel  to  AC,  will  represent  bend- 
ing moments  in  the  sections,  as  H,  at  their  feet. 


The  shear  at  any  section  is 


dM 

dx 


S  =  '-^~  =  R  -  fix  =  D  ( -  -  x  \      .    .     .    (7) 


or,  it  is  equal  to  the  load  covering  that  portion  of  the  beam  be- 
tween the  section  in  question  and  the  centre. 

Eq.  (7)  shows  that  the  shear  at  the  centre  is  zero ;  it  also  shows 
that  5  =  R  at  the  ends  of  the  beam.  It  further  demonstrates 
that  the  shear  varies  directly  as  the  distance  from  the  centre. 
Hence,  take  RB  to  represent  R  and  draw  BC.  The  shear  at 
any  section,  as  H,  will  then  be  represented  by  the  vertical  in- 
tercept, as  HG,  included  between  BC  and  RC. 

The  shear  being  zero  at  the  centre,  the  greatest  bending 
moment  will  also  be  found  at  that  point.  This  is  also  evident 
from  inspection  of  the  loading. 

Eq.  (2)  of  Case  I.,  shows  that  if  a  beam  of  span  /  carries  a 

W 
weight  —  at  its  centre,  the  moment  M  at  the  same  point  will 

be: 


The  third  member  of  Eq.  (8)  is  identical  with  the  third 
member  of  Eq.  (5).  It  is  shown,  therefore,  that  a  load,  concen- 
trated at  the  centre  of  a  non-continuous  beam,  will  cause  the  same 
moment,  at  that  centre,  as  double  the  same  load  uniformly  dis- 
tributed over  the  span. 

Eqs.  (5)  and  (8)  are  much  used  in  connection  with  the  bend- 
ing of  ordinary  non-continuous  beams,  whether  solid  or  flanged  : 
and  such  beams  are  frequently  found. 


142 


THEORY  OF  FLEXURE. 


[Art.  21. 


Case  Iff. 

The  third  case  to  be  taken,  is  a  cantilever  uniformly  loaded  ; 
it  is  shown  in  Fig.  3.     Let  x  and  xl  be 
D  measured  from  the  free  end  A,  and  let 

the  uniform  intensity  of  the  load  be 
represented  by  /.  The  entire  loading 
is  uniformly  continuous.  Hence  the 
principles  and  formulae  of  Art.  20  give, 
Fig.3  for  the  moment  about  any  section  x  : 


M=  - 


(9) 


If  AB  —  /,  the  moment  at  B  is  : 


(10) 


The  negative  sign  is  used  to  indicate  that  the  lower  side  of 
the  beam  is  subjected  to  compression.  In  the  two  preceding 
cases,  evidently,  the  upper  side  is  in  compression. 

The  shear  at  any  section  is  : 


dM 


ii) 


Hence,  the  shear  at  any  section  is  the  load  between  the  free 
end  and  that  section. 

Eq.  (9)  shows  that  the  moments  vary  as  the  square  of  the 
distance  from  the  free  end  ;  consequently,  the  moment  curve 
is  a  parabola  with  the  vertex  at  A,  and  with  a  vertical  axis. 
Let  BCy  then,  represent  Mt  by  any  convenient  scale,  and  draw 


Art.  22.] 


GENERAL  FLEXURE  FORMULAE. 


143 


the  parabola  CD  A.     Any  vertical  intercept  as  DF  will  repre- 
sent the  moment  at  the  section,  as  F,  at  its  foot. 

Again,  let  BG  represent  the  shear//,  at  B,  then  draw  the 
straight  line  AG.  Any  vertical  intercept,  as  HFy  will  then 
represent  the  shear  at  the  corresponding  section  F. 


Art.  22. — Recapitulation  of  the   General   Formulae  of  the  Common 
Theory  of  Flexure. 

It  is  convenient  for  many  purposes  to  arrange  the  formulae 
of  the  Common  Theory  of  Flexure  in  the  most  general  and 
concise  form.  In  this  Art.  the  preceding  general  formulas  for 
shears,  strains,  resisting  moments  and  deflections  will  be  re- 
capitulated and  so  arranged.  In  order  to  complete  the  gener- 
alization, the  summation  sign  2  will  be  used  instead  of  the 
sign  of  integration. 


In  Fig.  i,  let  ABC  represent  the  centre  line  of  any  bent 
beam  ;  AF,  a  vertical  line  through  A  ;  CF,  a  horizontal  line 
through  C,  while  A  is  the  section  of  the  beam  at  which  the 
deflection  (vertical  or  horizontal)  in  reference  to  C,  the  bend- 
ing moment,  the  shearing  stress,  etc.,  are  to  be  determined. 
As  shown  in  figure,  let  x  be  the  horizontal  co-ordinate  meas- 
ured from  A,  and y  the  vertical  one  measured  from  the  same 
point ;  then  let  z  be  the  horizontal  distance  from  the  same 


144  THEORY  OF  FLEXURE.  [Art.  22. 

point  to  the  point  of  application  of  any  external  vertical  force 
P.  To  complete  the  notation,  let  D.be  the  deflection  desired  ; 
Mv  the  moment  of  the  external  forces  about  A  ;  S,  the  shear 
at  A  ;  P',  the  strain  (extension  or  compression)  per  unit  of 
length  of  a  fibre  parallel  to  the  neutral  surface  and  situated  at 
a  normal  distance  of  unity  from  it  ;  /,  the  general  expression  of 
the  moment  of  inertia  of  a  normal  cross  section  of  the  beam, 
taken  in  reference  to  the  neutral  axis  of  that  section  ;  £,  the 
coefficient  of  elasticity  for  the  material  of  the  beam  ;  and  M 
the  moment  of  the  external  forces  for  any  section,  as  B. 

Again,  let  A  be  an  indefinitely  small  portion  of  any  normal 
cross  section  of  the  beam,  and  let  y'  be  an  ordinate  normal 
to  the  neutral  axis  of  the  same  section.  By  the  "  common 
theory  "  of  flexure,  the  intensity  of  stress  at  the  distance  y' 
from  the  neutral  surface  is  (y  'P  '£).  Consequently  the  stress 
developed  in  the  portion  J,  of  the  section,  is  EP'y'A,  and  the 
resisting  moment  of  that  stress  is  EP'y"2  A. 

The  resisting  moment  of  the  whole  section  will  therefore 
be  found  by  taking  the  sum  of  all  such  moments  for  its  whole 
area. 

Hence  : 

M  =  EP'2y'*A  =  EP'L 
Hence,  also  . 


El 


If  n  represents  an  indefinitely  short  portion  of  the  neutral 
surface,  the  strain  for  such  a  length  of  fibre  at  unit's  distance 
from  that  surface  will  be  nP'. 

If  the  beam  were  originally  straight  and  horizontal,  n  would 
be  equal  to  dx. 

P1  being  supposed  small,  the  effect  of  the  strain  nP'  at  any 


Art,  22.]  GENERAL  FLEXURE  FORMULAE.  145 

section,  By  is  to  cause  the  end  K  of  the  tangent  BK,  to  move 
vertically  through  the  distance  nP' x. 

If  BK  and  BR  (taken  equal)  are  the  positions  of  the  tan- 
gents before  and  after  flexure,  nP'x  will  be  the  vertical  dis- 
tance between  K  and  R. 

By  precisely  the  same  kinematical  principle,  the  expres- 
sion nP'y  will  be  the  horizontal  movement  of  A  in  reference 
to^. 

Let  2nP'x  and  2nP'y  represent  summations  extending 
from  A  to  £7,  then  will  those  expressions  be  the  vertical  and 
horizontal  deflections,  respectively,  of  A  in  reference  to  C.  It 
is  evident  that  these  operations  are  perfectly  general,  and  that 
x  and  y  may  be  taken  in  any  direction  whatever. 

The  following  general,  but  strictly  approximate  equations, 
relating  to  the  subject  of  flexure,  may  now  be  written  : 

S    =  2P (i) 

(2) 


(4) 

D  =  ?nP'x  =  S^f- (5) 

(6) 


Dh  represents  horizontal  deflection. 
10 


146  THEORY  OF  FLEXURE.  [Art.  23. 


The  summation  ^Pz  must  extend  from  A  to  a  point  of  no 
bending  ;  or  from  A  to  a  point  at  which  the  bending  moment 
is  MJ.  In  the  latter  case  : 


+  MS  .......   (7) 

Ml  may  be  positive  or  negative. 

Art.  23.—  The  Theorem  of  Three  Moments. 

The  object  of  this  theorem  is  the  determination  of  the  re- 
lation existing  between  the  bending  moments  which  are  found 
in  any  continuous  beam  at  any  three  adjacent  points  of  sup- 
port. In  the  most  general  case  to  which  the  theorem  applies, 
the  section  of  the  beam  is  supposed  to  be  variable,  the  points 
of  aiipport  are  not  supposed  to  be  in  the  same  level,  and  at 
any  jppiat,  or  all  points,  of  support  there  may  be  constraint 
applied  to  the  beam  external  to  the  load  which  it  is  to  carry  ; 
or,  what  is  equivalent  to  the  last  condition,  the  beam  may  not 
be  straight  at  any  point  of  support  before  flexure  takes  place. 

Before  establishing  the  theorem  itself,  some  preliminary 
matters  must  receive  attention. 

If  a  beam  is  simply  supported  at  each  end,  the  reactions 
are  found  by  dividing  the  applied  loads  according  to  the 
simple  principle  of  the  lever.  If,  however,  either  or  both  ends 
are  not  simply  supported,  the  reaction,  in  general,  is  greater  at 
one  end  and  less  at  the  ottyer  than  would  be  found  by  the  law 
of  the  lever  ;  a  portion  of  the  reaction  at  one  end  is,  as  it  were, 
transferred  to  the  other.  The  transference  can  only  be  ac- 
complished by  the  application  of  a  couple  to  the  beam,  the 
forces  of  the  couple  being  applied  at  the  two  adjacent  points 
of  support  ;  the  span,  consequently,  will  be  the  lever  arm  of 
the  couple.  The  existence  of  equilibrium  requires  the  .appli- 


Art.  23.]  THEOREM  OF   THREE  MOMENTS. 


147 


cation  to  the  beam  of  an  equal  and  opposite  couple.  It  is  only 
necessary,  however,  to  consider,  in  connection  with  the  span 
AB,  the  one  shown  in  Fig.  i.  Further,  from  what  has  imme- 
diately preceded,  it  appears  that  the  force  of  this  couple  is 


equal  to  the  difference  between  the  actual  reaction  at  either 
point  of  support  and  that  found  by  the  law  of  the  lever.  The 
bending  caused  by  this  couple  will  evidently  be  of  an  opposite 
kind  to  that  existing  in  a  beam  simply  supported  at  each  end. 
These  results  are  represented  graphically  in  Fig.  I.  A  and 
B  are  points  of  support,  and  AB  is  the  beam  ;  AR  and  BR 
are  the  reactions  according  to  the  law  of  the  lever ;  RF  =  R'F 
is  the  force  of  the  applied  couple ;  consequently  : 

AF  =  AR  +  RF    and    BF  =  BR'  -  (R'F  =  RF) 


are  the  reactions  after  the  couple  is  applied.  As  is  well  known, 
lines  parallel  to  CK,  drawn  in  the  triangle  ACB,  represent  the 
bending  moments  at  the  various  sections  of  the  beam,  when 
the  reactions  are  AR  and  BR.  Finally,  vertical  lines  parallel 
to  AG,  in  the  triangle  QHG,  will  represent  the  bending  mo- 
ments caused  by  the  force  R'F. 


14^  THEORY  OF  FLEXURE.  [Art.  23. 

In  the  general  case  there  may  also  be  applied  to  the  beam 
two  equal  and  opposite  couples,  having  axes  passing  through 
A  and  B  respectively.  The  effect  of  such  couples  will  be 
nothing  so  far  as  the  reactions  are  concerned,  but  they  will 
cause  uniform  bending  between  A  and  B.  This  uniform  or 
constant  moment  may  be  represented  by  vertical  lines  drawn 
parallel  to  AH  or  LN  (equal  to  each  other)  between  the  lines 
AB  and  HQ.  The  resultant  moments  to  which  the  various 
sections  of  the  beam  are  subjected  will  then  be  represented  by 
the  algebraic  sum  of  the  three  vertical  ordinates  included  be- 
tween the  lines  ACB  and  GQ.  Let  that  resultant  be  called  M. 

Let  the  moment  GA  be  called  Ma,  and  the  moment  : 

BQ  =  LN  =  HA  =  Mb. 

Also  designate  the  moment  caused  by  the  load  P,  shown  by 
lines  parallel  to  CK  in  A  CB,  by  Mv  Then  let  x  be  any  hori- 
zontal distance  measured  from  A  toward  B  ;  /  the  horizontal 
distance  AB ;  and  z  the  distance  of  the  point  of  application, 
K,  of  the  force  P  from  A.  With  this  notation  there  can  be  at 
once  written  :' 


M  =  ^.(— — +  M,-  }+Mt.    .    .    .    (i) 


Eq.  (i)  is  simply  the  general  form  of  Eq.  (2). 

It  is  to  be  noticed  that  Fig.  i  does  not  show  all  the  mo- 
ments MM  Mb  and  Mt  to  be  of  the  same  sign,  but,  for  conven- 
ience, they  are  so  written  in  Eq.  (i). 

The  formula  which  represents  the  theorem  of  three  mo- 
ments can  now  be  written  without  difficulty.  The  method  to 
be  followed  involves  the  improvements  added  by  Prof.  H.  T. 
Eddy,  and  is  the  same  as  that  given  by  him  in  the  "American 
Journal  of  Mathematics,"  Vol.  I.,  No.  i. 


149 


Art.  23.]  THEOREM  OF   THREE  MOMENTS. 

Fig.  2  shows  a  portion  of  a  continuous  beam,  including  two 
spans  and  three  points  of  support.  The  deflections  will  be 
supposed  measured  from  the  horizontal  line  NQ.  The  spans 


N 

J-_ 

cf 

fll 

u                              /T<- 

!°. 

"1cc 

j^.  __£'___  4, 

1 

!M« 

JMt 

t       . 

?Us* 

fUs- 

Fig.  2 


are  represented  by  4  and  lc ;  the  vertical  distances  of  NQ  from 
the  points  of  support  by  caj  cb  and  cc ;  the  moments  at  the  same 
points  by  Ma1  Mb  and  Mc,  while  the  letters  5  and  R  represent 
shears  and  reactions  respectively. 

In  order  to  make  the  case  general,  it  will  be  supposed  that 
the  beam  is  curved  in  a  vertical  plane,  and  has  an  elbow  at  b, 
before  flexure,  and  that,  at  that  point  of  support,  the  tangent 
of  its  inclination  to  a  horizontal  line,  toward  the  span  la  is  /, 
while  t'  represents  the  tangent  on  the  other  side  of  the  same 
point  of  support ;  also  let  d  and  d'  be  the  vertical  distances, 
before  bending  takes  place,  of  the  points  a  and  c,  respectively, 
below  the  tangents  at  the  point  b. 

A  portion  of  the  difference  between  ca  and  cb  is  due  to  the 
original  inclination,  whose  tangent  is  /,  and  the  original  lack  of 
straightness,  and  is  not  caused  by  the  bending ;  that  portion 
which  is  due  to  the  bending,  however,  is,  remembering  Eq. 
(5),  Art.  22  : 

«  Mxn 

D  =  ca  -  cb  -  lat  -  d  =  2b  --- . 


15°  THEORY  OF  FLEXURE.  [Art.  23. 

By  the  aid  of  Eq.  (i)  this  equation  may  be  written  : 

E(ca-  cb  -  lat  -  d) 


In  this  equation,  it  is  to  be  remembered,  both  x  and  z  (in- 
volved in  M^  are  measured  from  support  a  toward  support  b. 
Now  let  a  similar  equation  be  written  for  the  span  /c,  in  which 
the  variables  x  and  z  will  be  measured  from  c  toward  b.  There 
will  then  result  : 

E(cc-  cb-  //  -  d1) 


When  the  general  sign  of  summation  is  displaced  by  the 
integral  sign,  n  becomes  the  differential  of  the  axis  of  the 
beam,  or  ds.  But  ds  may  be  represented  by  u  dx,  u  being  such 
a  function  of  x  as  becomes  unity  if  the  axis  of  the  beam  is 
originally  straight  and  parallel  to  the  axis  of  x.  The  Eqs.  (2) 
and  (3)  may  then  be  reduced  to  simpler  forms  by  the  following 
methods  : 

In  Eq.  (2)  put  : 

y»  (I  —  x\  xn         i  (a  u  (/a  —  x)x  dx 

'\r-J~r  '-  7jb~    ~r~ 

>aJbl(la~l*}dX (4) 

Also: 


Art.  23.]  THEOREM  OF   THREE  MOMENTS.  151 


b 


Also  : 

l.-*)d*=*  (6) 


In  the  same  manner  : 

«  ^r2^         I  f  a  «*2  ^        ^a  f  a  uxdx  ,  N 

^7aT==7j,~7-     :  7J»~/~ 

Also: 

^r^=^f^^     ......     (8) 

4  J  *         ^  *  a     -1  4 

And, 

jMfe.    ...    (9) 


Again,  in  the  same  manner  : 

....    (10) 


Using  Eqs.  (4)  to  (10),  Eq.  (2)  may  be  written 

-£  (Cc  Cb  I  of   ~  '    ~  ~    2    ^       a    a  a    a 

Ax 


IS2  THEORY  OF  FLEXURE.  [Art.  23. 

Proceeding  in  precisely  the  same  manner  with  the  span  ln 
Eq.  (3)  becomes  : 


E(cc  -  cb-  //  -  d'}  = 


x 


(12) 


The  quantities  xa  and  xc  are  to  be  determined  by  applying 
Eq.  (4)  to  the  span  indicated  by  the  subscript ;  while  ua,  ia,  uc 
and  ie  are  to  be  determined  by  using  Eqs.  (5)  and  (6)  in  the 
same  way.  Similar  observations  apply  to  u'a,  i'a,  x'M  uc,  i'c  and 
x'c,  taken  in  connection  with  Eqs.  (7),  (8)  and  (9). 

If  7  is  not  a  continuous  function  of  x,  the  various  integra- 
tions of  Eqs.  (4),  (5),  (7)  and  (8)  must  give  place  to  summations 
(2)  taken  between  the  proper  limits. 

Dividing  Eqs.  (11)  and  (12)  by  la  and  lct  respectively,  and 
adding  the  results  : 

R  (Ca  ~  Cb  4-  Ce  ~  Cb         T  -d  -     l 

i         ~r~  7 

ia  LC  LO, 


x 


''a 

iaxa  +  Mbuai'ax'a  +  M&jisc  +  Mbuci'cx^  .     (13) 


in  which  T  =/  +  /'. 

Eq.  (13)  is  the  most  general  form  of  the  theorem  of  three 
moments  if  E,  the  coefficient  of  elasticity,  is  a  constant  quan- 
tity. Indeed,  that  equation  expresses,  as  it  stands,  the  "  the- 
orem "  for  a  variable  coefficient  of  elasticity  if  (ie)  be  written 
instead  of  i\  e  representing  a  quantity  determined  in  a  manner 
exactly  similar  to  that  used  in  connection  with  the  quantity  i. 


Art.  23.]  THEOREM  OF   THREE  MOMENTS.  153 

In  the  ordinary  case  of  an  engineer's  experience  T  =  o, 
d  —  d '  —  o,  /  =  constant,  u  =  i^  =  uc  —  etc.,  =  c  =  secant  of 
the  inclination  for  which  t  —  —  t'  is  the  tangent ;  consequently  : 


From  Eq.  (4)  : 


From  Eq.  (7)  : 


- 

6    ' 


The  summation  SM^  Ax  can  be  readily  made  by  referring 
to  Fig.  i. 

The  moment  represented  by  CK  in  that  figure  is  : 


consequently  the  moment   at  any  point   between  A  and  K, 
due  to  P,  is  : 


Between  K  and  B  : 


154  THEORY  OF  FLEXURE.  [Art.  23. 

Using  these  quantities  for  the  span  la  : 

2*bMs  Ax  =  [*  MS  dx  +  \M±x  dx  =  Yf>P(ll  -  *2)z. 

Jo  *  z 

For  the  span  lc,  the  subscript  a  is  to  be  changed  to  c. 
Introducing    all  these    quantities    Eq.  (13)  becomes,  after 
providing  for  any  number  of  weights,  P\ 


C 


(4 


+  1  2p(i:  -  <>•+'•  r  2p(if  -  *•>»  .  ..  .   (14) 

*a  *c 


Eq.  (14),  with  c  equal  to  unity,  is  the  form  in  which  the 
theorem  of  three  moments  is  usually  given  ;  with  c'  equal  to 
unity  or  not,  it  applies  only  to  a  beam  which  is  straight  before 
flexure,  since  : 

T-  t  +  t'  =  o  =  d=  d'. 

If  such  a  beam  rests  on  the  supports  a,  b,  and  c,  before 
bending  takes  place, 


4  4 

and  the  first  member  of  Eq.  (14)  becomes  zero. 

If,  in  the  general  case  to  which  Eq.  (13)  applies,  the  deflec- 
tions CM  Ci,  and  cc  belong  to  the  beam  in  a  position  of  no  bend- 
ing, the  first  member  of  that  equation  disappears,  since  it  is 
the  sum  of  the  deflections  due  to  bending  only,  for  the  spans  /a, 
and  1C9  divided  by  those  spans,  and  each  of  those  quantities  is 


Art.  23.]  THEOREM  OF   THREE  MOMENTS.  155 

zero  by  the  equation  immediately  preceding,  Eq.  (2).  Also,  if 
the  beam  or  truss  belonging  to  each  span  is  straight  between 
the  points  of  support  (such  points  being  supposed  in  the  same 
level  or  not),  ua  —  u'a  ~  ula  =  constant,  and  uc  =  uc  =  uu  =  an- 
other constant.  If,  finally,  /be  again  taken  as  constant,  xa  and 
xc,  as  well  as  2M^r  Ax,  will  have  the  values  found  above. 

From  these  considerations  it  at  once  follows  that  the  second 
member  of  Eq.  (14),  put  equal  to  zero,  expresses  the  theorem 
of  three  moments  for  a  beam  or  truss  straight  between  points 
of  support,  when  those  points  are  not  in  the  same  level,  but 
when  they  belong  to  a  configuration  of  no  bending  in  the 
beam.  Such  an  equation,  however,  does  not  belong  to  a  beam 
not  straight  between  points  of  support. 

The  shear  at  either  end  of  any  span,  as  /a,  is  the  next  to  be 
found,  and  it  can  be  at  once  written  by  referring  to  the  obser- 
vations made  in  connection  with  Fig.  I.  It  was  there  seen 
that  the  reaction  found  by  the  simple  law  of  the  lever  is  to  be 
increased  or  decreased  for  the  continuous  beam,  by  an  amount 
found  by  dividing  the  difference  of  the  moments  at  the  ex- 
tremities of  any  span  by  the  span  itself.  Referring  therefore, 
to  Fig.  2,  for  the  shears  S,  there  may  at  once  be  written  : 


V 


(16) 
(-7) 


156  THEORY  OF  FLEXURE.  [Art.  23. 

The  negative  sign  is  put  before  the  fraction, 

Ma  -  Mb 
4 

in  Eq.  (15),  because  in  Fig.  I  the  moments  Ma  and  Mb  are  rep- 
resented opposite  in  sign  to  that  caused  by  P,  while  in  Eq.  (i) 
the  three  moments  are  given  the  same  sign,  as  has  already 
been  noticed. 

Eqs.  (15)  to  (18)  are  so  written  as  to  make  an  upward  re- 
action positive,  and  they  may,  perhaps,  be  more  simply  found 
by  taking  moments  about  either  end  of  a  span.  For  example, 
taking  moments  about  the  right  end  of  la : 

SJ.  -  IP  (4  -*)  +  M.=  Mb. 

From  this,  Eq.  (15)  at  once  results.  Again,  moments  about 
the  left  end  of  the  same  span  give  : 

5,7.  -  2Ps  +  Mt  =  Ma. 


This  equation  gives  Eq.  (16),  and  the  same  process  will  give 
the  others. 

If  the  loading  over  the  different  spans  is  of  uniform  inten- 
sity, then,  in  general,  P=  w  dz  \  w  being  the  intensity.  Con- 
sequently : 

fl  74 

2P(/2  -  z2}z  =       w  (I2  -  z2)  zdz  =  w  —  . 
J0  4 

In  all  equations,  therefore,  for 


Art.  230.]  CONTINUOUS   REACTIONS.  157 

there  is  to  be  placed  the  term  wa  -^  ;  and  for 

4 

i     c 
lc 

/3 

the  term  wc  -~.     The   letters  a  and  c  mean,   of  course,  that 

reference  is  made  to  the  spans  la  and  lc. 

From  Fig.  2,  there  may  at  once  be  written  : 

R    =  S;  -f  Sa (19) 

R'  =Si  +  St (20) 

R"  =  S'c  +  Sc (21) 

etc.  =  etc.  -f-  etc. 

Art.  23a. — Reactions  under  Continuous  Beam  of  any  Number  of  Spans. 

The  general  value  of  the  reactions  at  the  points  of  support 
under  any  continuous  beam  have  been  given  in  Eqs.  (19),  (20), 
(21),  etc.,  of  the  preceding  Art.  Before  those  equations,  how- 
ever, can  be  applied  to  any  particular  case,  the  values  of  the 
bending  moments,  which  appear  in  the  expressions  Sa,  S^,  S&, 
etc.,  for  the  shears,  must  be  determined.  In  the  application  of 
the  theorem  of  three  moments,  it  is  invariably  virtually  as- 
sumed that  the  continuous  beam  before  flexure  is  straight 
between  the  points  of  support,  and  that  the  latter  belong  to  a 
configuration  of  no  bending.  The  moment  of  inertia  7  and 
the  coefficient  of  elasticity  E  are  also  assumed  to  be  constant. 
This  is  frequently  not  strictly  true,  yet  it  will  be  assumed  in 


158  THEORY  OF  FLEXURE.  '  [Art. 

what  follows,  since  the  method  to  be  used  in  finding  the  mo- 
ments is  entirely  independent  of  the  assumption,  and  remains 
precisely  the  same  whatever  form  for  the  theorem  of  three 
moments  may  be  chosen. 

Agreeably  to  the  assumption  made,  Eq.  (14)  of  the  preced- 
ing Art.  takes  the  following  form,  which  is  almost,  or  quite, 
invariably  used  in  engineering  practice  : 


MJa  +  2Mb(la  +  /,)  +  MJ<=-   -f 

*a 


(i) 


Let  Fig.  i  represent  a  continuous  beam  of  n  spans,  equal 
or  unequal  in  length.     At  the  points  of  support,  o,  1,2,  3,  4,  5, 

p        A  fa  h  I*  h      ^         I*      A 

Fig.1 


etc.,  let  the  bending  moments  be  represented  by  Mw  M»  M^ 
My  etc.  The  moment  MQ  is  always  known  ;  it  is  ordinarily 
zero,  and  that  will  be  considered  its  value. 

An  examination  of  Fig.  i  shows  that,  by  repeated  applica- 
tions of  Eq.  (i),  the  number  of  resulting  equations  of  condition 
will  be  one  less  than  the  number  of  spans.  But  if  the  two  end 
moments  are  known  (here  assumed  to  be  zero),  the  number  of 
unknown  moments  will  also  be  one  less  than  the  number  of 
spans.  Hence  the  number  of  equations  will  always  be  suffi- 
cient for  the  determination  of  the  unknown  moments. 

For  the  sake  of  brevity  let  the  following  notation  be 
adopted  : 


Art.  230.]  CONTINUOUS  REACTIONS.  159 


-7 

'4 


etc.  =  etc.  —  etc. 

a,  =  2(1,  +  /,) ;  V  =  /, 


A  = 


/  denoting  any  number  of  the  series  I,  2,  3,  4,  ......  ;/.     It  is 

thus  seen  that,  in  general, 

ql  =  2(pt  +  Sf)  ; 

also  that  a2  =  bv  c2  =  by  d^  =  c^  etc.     These  relations  can  be 
used  to  simplify  the  final  result. 

By  repeated  applications  of  Eq.  (i)  the  following  ;/  equa- 
tions of  condition,  involving  the  notation  given  above,  will 
result  : 


i6o 


THEORY  OF  FLEXUKE. 


[Art. 


b2M2 


(2) 


The  moment  J/M  +  Iwill  also  be  equal  to  zero.  In  conse- 
quence of  this  last  condition  it  is  seen  that  the  coefficients  of 
the  Ms  occupy  precisely  the  places  of  the  elements  of  a  deter- 
minant of  the  72th  degree.  Of  the  array  indicating  the  deter- 
minant, however,  there  exists  only  the  leading  diagonal  and 
one  diagonal  on  each  side  of  it.  The  determinant  for  n  equa- 
tions, or  (n  +  i)  spans,  has,  then,  the  value  : 


#,,#,,0,0,0,0,.      .       .       .       . 

a»  b2,  c2J  o  ,  o ,  o , 


O     ,       by       Cy      dy      O    ,       O     , 

o  ,  o  ,  r 4,  d^  /4,  o 

0,0,0,       dyfygy 


•     •     •     (3) 


o,  0,0,0,0,     .     .     .     .     o,  /„,  qn 
Also  let  Df  represent  the  value  of  the  determinant  D  when 


Art.  23*.] 


CONTINUOUS  REACTIONS. 


161 


the  column  indicated  by  the  zth  letter  of  the  series  a,  b,  c,  d,  /, 
etc.,  is  replaced  by  the  column  «x,  u3,  Uy  u4,  etc.  If,  for  exam- 
ple, i  =  3,  the  2th  letter  is  c.  Hence  : 

ai>  &D  u»  °  >  O  >  O  , 

<72,     #2,     Z/2,    O   ,    O   ,    O   , 

o  ,  o ,  u4,  d^  /4,  o , .    -    •    (4) 

0,0,      Uy      dy    fy    gy 

•  •  •  •  . 

i  O  ,  O  ,  f/w,  O  ,  O  ,  O  ,  .      .     0,pnJqn    j 
Then,  in  general  : 

"•=% (5) 

Eq.  (5)  will  give  the  value  of  the  bending  moment  at  any 
point  of  support,  whatever  may  be  the  number  of  spans  or  the 
law  of  loading  on  any  or  all  the  spans. 

Precisely  the  same  formulae  are  to  be  used  if  M0  and  Mn  are 
not  zero,  but  have  definite  values  and  are  known.  In  such  a 
case,  however,  ut  and  UH  would  be  replaced  by  : 

«;  =  us  -  a0M0. 


The  same  equations  also  hold  true  whatever  form  of  the 
theorem  of  three  moments  may  be  chosen.     It  is  only  to  be 
IT 


l62  THEORY  OF  FLEXURE.  [Art. 

remembered  that  the  values  of  the  quantities  a,  b,  c,  etc.,  uv 
uv  uy  etc.,  will  depend  upon  the  choice. 

If  all  the  moments  are  desired,  it  will  be  most  convenient 
to  put  the  vertical  column  #x,  ual  ny  .  .  .  un  in  place  of  the  ver- 
tical column  <?!,  a2,  o,  o,  .  .  .  o,  in  Eq.  (4),  and  then  find  the 
resulting  determinant  D^.  Eq.  (5)  will  then  give  the  value  of 
Mv  which,  placed  in  the  first  of  Eqs.  (2),  will  enable  M2  to  be 
at  once  found.  M3  will  then  result  from  the  second  of  Eqs.  (2), 
M4  from  the  third,  etc.,  etc. 

So  far  as  the  general  treatment  of  the  question  is  con- 
cerned, there  yet  remains  to  be  considered  the  expansion  of 
the  determinants  D  and  Z^. 

The  expansion  of  the  determinant  D  is  very  simple,  and 
leads  to  the  following  results  : 

For  two  spans  : 

D  =  a,     ........     (6) 

For  three  spans  : 

D  =  aA  -  aA  ...     v  ...     (7) 
For  four  spans  : 

D  =  aj}^  -  ajbjz  -  ajb^ (8) 

For  five  spans  : 
D  =  aA^4  -  ajbf^  -  a&cjt4  -  aj>j&  +  ajf^dz\     (9) 

For  six  spans  : 

D  = 


Art.  230.]  CONTINUOUS  REACTIONS.  163 

By  the  observance  of  two  or  three  simple  rules,  the  deter- 
minant  for  (n  -f-  i)  spans,  or  n  points  of  support,  may  easily  be 
written. 

A  series  of  numbers  such  as  I,  2,  3,  4,  5,  6,  etc.,  is  said  to 
be  written  in  its  natural  order.  Let  any  permutation  of  this 
series,  2,  I,  3,  6,  5,  4,  be  written,  in  which  2  is  placed  before  I, 
6  before  5  and  4,  and  5  before  4.  In  this  permutation,  there- 
fore, there  are  said  to  be  (i  -f  2  -f-  i)  =  4  inversions. 

Let  (Aw)  represent  any  letter  of  the  series  a,  b,  c,  d,  etc.,  which 
has  the  subscript  n  ;  also,  let  (Xn)n  and  (AW)M  _  x  represent  the  nth 
and  (n  —  i)th  letters  of  the  same  series  which  have  the  sub- 
scripts n.  In  general,  the  letter  inside  the  parenthesis  repre- 
sents the  subscript  figure  in  the  determinant,  and  that  outside, 
the  place  of  the  letter  in  the  series  a,  b,  c,  d,  f,  etc. 

The  nth  determinant  for  (n  -j-  i)  spans,  or  n  points  of  sup- 
port, will  then  be  : 


Now,  with  the  notation  taken,  if  the  letters  in  each  term  of 
the  determinant  are  written  in  their  natural  order,  as  abcdfg, 
etc.,  the  number  of  inversions  in  the  subscript  figures  of  any  term 
will  determine  the  sign  of  that  term,  i.e.,  if  the  number  of  in- 
versions is  odd,  the  sign  is  minus,  but  if  the  number  is  even  the 
sign  is  plus. 

Since  n  is  the  greatest  subscript  in  any  term,  and  since  (AH)n 
occupies  the  most  advanced  place  in  the  series  of  letters,  no 
inversions  are  introduced  in  multiplying  DM_l  by  (AW)M.  Hence, 
all  terms  of  Dn  _  x  (\H)H  will  have  the  same  signs  as  the  correspond- 
ing terms  of  Dn  _  x  . 

Similarly,  since  n  is  greater  than  (n  —  i),  the  product 
(A«)«  -  1  0«  -  x)«  involves  one  inversion.  Hence,  all  terms  of 


164 


THEORY  OF  FLEXURE. 


[Art.  230. 


will  have  signs  contrary  to  those  of  the  corresponding  terms  of 

/>..,: 

The  number  of  terms  in  Dn  will  evidently  be  the  sum  of 
the  numbers  of  terms  in  Dn  _  x  and  Dn_2. 

An  examination  of  the  notation  will  at  once  show  that  : 

0»)«  =  2(4  +  4  +  x)  ;    (*„)«  -  x  -  4  ;    and    (AM  _  ,)„  =  4. 

Hence  there  will  result  : 


The  minus  sign  before  the  last  term  of  the  second  member 
is  on  account  of  the  inversion  introduced,  as  already  ex- 
plained. 

The  general  value  of  the  determinant  D£  (shown  in  Eq.  (4) 
when  i  —  3)  can  be  most  easily  expanded  by  considering  it 
the  sum  of  two  determinants  ;  and  in  order  to  illustrate  this 
method  let  it  be  supposed  that  M3  is  desired.  It  will  then  be 
necessary  to  expand  the  determinant  Dy  given  in  Eq.  (4).  As 
is  known  from  the  theory  of  determinants,  Dz  may  be  written 
as  follows  : 


a»  bv 

0,0,0, 

0,  .    .   . 

'*„  b»  ^,0,0,0,  .  . 

a2J  b2, 

«2»    0,0, 

o  ,  .  .  . 

#a,  b2,  o  ,  o  ,  o  ,  o  ,  .  . 

o,  by 

Uy       dy       0    , 

0,  .    .    . 

o  ,  by  o  ,  dy  o  ,  o  ,  .  . 

0,  0, 

*4'4'/4» 

0,  .    .    . 

>    +    ' 

o  ,  o  ,  o  ,  d#  /4,  o  ,  .  . 

.(12) 

o,  o, 

0     ,       dy    fy 

gy  '   •    • 

0       ,          0      ,          Uy          dy      fy       gy             .              . 

.0,0,0 

0,  0,  0,  0, 

.    AM    <7«- 

0,  0,  «„,  0,  0,  0,  .  .  At,  <7« 

Art.  230.]  CONTINUOUS  REACTIONS.  165 


or  : 


Eq.  (12)  shows  at  a  glance  what  D±  and  Z>3"  represent. 

Z>3'  is  precisely  the  same  in  form  as  D,  and  is  given  at  once 
by  the  Eqs.  (6)  to  (11)  after  writing  ua,  uz  and  u4  for  ca,  c3 
and  cv 

In  general,  D-  is  found  by  simply  writing  «,._„  #,.  and  #,-  +  , 
for  (A,.  _,),.,  (At.)t-  and  (A,-  +  I)  »•  in  the  determinant  A 

As  a  general  method,  that  of  alternate  numbers  is  probably 
as  simple  as  any  for  the  expansion  of  the  determinant  £>•'. 
For  example  : 


A"  =  («.'x  +  V2  +  v 

.  .  .  («^s+A^-i  +  ^«);  .....    (14) 

in  which  et,  ea,  ey  etc.,  are  the  units  of  the  alternate  numbers. 

The  circumstances  of  any  particular  case  will  frequently 
either  furnish  a  more  expeditious  method  than  that  of  alternate 
numbers,  or  allow  the  expansion  of  D-'  to  be  written  at  once 
from  an  inspection  of  the  array  given  in  Eq.  (12). 

In  any  case  the  method  of  alternate  numbers  may  be  used 
as  a  check. 

Special  Method  for  Ordinary  Use. 

If  the  number  of  spans  is  large,  the  expansion  of  the  deter- 
minant Di  will,  at  best,  be  found  somewhat  tedious.  Special 
methods  may  be  employed  which  involve  only  the  determinant 
Z>,  given  in  Eqs.  (6)  to  (11)  ;  and  it  has  already  been  seen  that 
that  determinant  admits  of  a  very  simple  expansion. 

Let  any  one  span  carry  any  load  whatever,  while  all  other 
spans  carry  no  load.  In  such  a  case,  P  will  be  zero  for  every 


1 66  THEORY  OF  FLEXURE.  [Art. 

span  but  one,  and,  in  consequence  of  the  notation  employed, 
all  but  two  quantities  in  the  series  ul}  u2y  uy  u4,  uy  etc.,  will  also 
become  equal  to  zero. 

If  4  (the  zth  span)  carries  the  load,  there  will  result  : 


- 

'» 


(15) 
(i6) 


All  other  «s  reduce  to  zero.  Although  Eqs.  (15)  and  (16) 
have  the  same  form,  they  are  not  identical  except  in  special 
cases,  since  z  is  not  measured  from  the  same  end  of  the  span 
in  both  expressions. 

Now  let  Ui_t  and  ut  take  the  place  of  those  letters  in  that 
column  of  D  formed  with  the  zth  letter  of  the  series  a,  b,  c,  d, 
etc.,  which  have  the  subscripts  i  and  i  —  I  ;  «i  +  I-is  equal  to 
zero.  Or  in  the  notation  already  employed,  let  ^ii_l  and  ^ 
take  the  place  of  (A,-.,),-  and  (A^),-,  while  zero  takes  the  place  of 
(A<  +  J),-.  The  resulting  determinant,  Diy  will  then  be  precisely 
the  same  as  D  in  general  form.  The  expansion  of  D{  can  then 
be  at  once  made  by  simply  putting  in  D  the  substitutions 
above  indicated.  There  will  then  result  : 


In  order  to  find  M{.lt  with  the  same  loading  on  the  same 
span,  «,-_,  and  uf  must  take  the  place  of  (A,- _,),-_,  and  (A;)t._x,  re- 
spectively, while  (A,-.^.,  becomes  equal  to  zero.  Making 
these  substitutions  in  the  determinant  D,  there  will  result  the 
determinant  -  Then  : 


Art.  23«.]  CONTINUOUS  REACTIONS.  167 


08) 


The  values  of  M£  and  ^_x,  thus  obtained,  placed  in  the  zth 
and  (i  —  i)th  of  the  Eqs.  (2)  will  at  once  give  : 


and 


Similar  substitutions  in  the  other  equations  will  give  all  the 
moments.  Thus  the  solution  is  complete,  for  the  span  and 
loading  taken,  with  the  use  of  the  expanded  determinant  D 
only. 

Each  span  may  be  treated  in  the  same  manner  and  the 
same  expansion  of  D  will  be  the  only  one  necessary. 

This  method  is  equivalent  to  splitting  the  elements  //„  ua, 
uy  u4,  etc.,  of  the  general  determinant  Dt. 

In  order  to  determine  the  bending  moment  at  any  point  of 
support,  for  loading  which  covers  more  than  one  span,  or  por- 
tions of  more  than  one  span,  it  is  only  necessary  to  take  the 
algebraic  sum  of  the  separate  moments  (as  above  determined), 
at  the  point  of  support  in  question,  found  for  the  loading  in 
each  single  span.  The  result  will  be  the  moment  due  to  the 
combined  action  of  all  the  loading. 

It  is  thus  seen  that  the  solution  of  the  most  general  case  is 
made  to  depend  on  the  one  expansion  of  the  determinant  D. 


Example. 

Let  there  be  a  continuous  beam  of  six  spans,  and  let  any 
loading  rest  upon  the  fourth  ;  it  is  required  to  find  the  expan- 
sions of  the  determinants  Dt  and  A-i- 

The  expansion  of  D  is  given  in  Eq.  (10),  and  need  not  be 
repeated  here. 


1  68  THEORY  OF  FLEXURE.  [Art.  230. 

Using  the  preceding  notation  : 

i          =4-  (*<-x)<  =  ^- 

1-1=3.  (A,),      =  d4. 

Ui         =  u4.  (A,.  +  ,)<  =  dy 


In  Eq.  (10)  then,  d4  and  dz  are  to  be  displaced  by  u4  and  #3, 
while  zero  is  to  take  the  place  of  dy     Hence  : 


Again  : 

(A,  _,),_,  =  ^3  (A,)._x  =  ^4. 

Then,  in  Eq.  (10),  placing  z/3  and  u4  for  <r3  and  c4,  and  placing 


there  will  result  : 

A  -  I    =    «  A' 


These  values  placed  in  Eqs.  (17)  and  (18)  will  give  J/4 
and  J/3. 

The  lengths  of  span  may  be  any  whatever  ;  .if  they  are  equal, 
the  results  will  be  simplified. 


Art.  230.]  CONTINUOUS  REACTIONS.  169 


Special  Case  of  Equal  Spans. 

If  all  the  spans  are  of  equal  length,  each  may  be  repre- 
sented by  /.     There  will  then  result  : 

a2  =  b  3  =  r4  =  4  =...=/,  =  £x  =  <ra  =  4  =/4=  •  ..*<=/) 

h     (21) 


These  values  of  #,  £,  £,  etc.,  placed  in  Eqs.  (6)  to  (10)  give  : 

For  two  spans  : 

D  =4!. 

For  three  spans  : 

D  =  IS/2. 

For  four  spans  : 

D  =  56/3. 

For  five  spans  : 

D  —  200,/4. 

For  six  spans  : 

D  =  78o/5. 

Others  may  be  easily  and  rapidly  written  by  the  aid  of  Eq. 
(n),  which  now  becomes  : 


I-/2^_2     .....     (22) 
If  the  determinant  for  seven  (i.e.,  n  +  i)  spans  is  desired  : 


170  THEORY  OF  FLEXURE.  [Art. 

£>„_!  =  ;8o/5     and     £>n-2  =  2O9/4. 
Hence  : 

Dn  —  £>6  =  3i"20/6  —  209/6  =  291  1/6. 

Similarly  for  eight  spans  : 

D  =  4  x  2911/7  -  78o/7  =  10864/7. 
For  nine  spans  : 

D  =  4  x  io864/8  -  291  1/8  =  40545/8. 
For  ten  spans  : 

D  —  4  X  40545/9  —  10864/9  =  151316/9. 


The  values  given  in  Eq.  (21)  will  correspondingly  simplify 
the  expansion  of  the  determinant  Diy  either  in  its  general  form 
as  exemplified  in  Eq.  (4)  or  as  given  in  the  special  method. 
As  an  illustration,  Eqs.  (19)  and  (20)  become,  respectively  : 


%4  —  6o/4//3. 
A  -i  =  225/^3  —  6o/4#4. 
These  values  then  give  : 

IP 

M  -  A  _ 

"*  -  D  - 


Art.  24.]  THE  NEUTRAL   CURVE.  i;i 

Then  by  Eqs.  (2)  :' 


Thus  all  the  moments  are  known  for  this  example,  t.e,  with 
six  spans  and  loading  on  the  fourth  span  only. 


Reactions. 

After  the  moments  are  found  either  by  the  general  or 
special  method,  for  any  condition  of  loading,  the  reactions  will 
at  once  result  from  the  substitution  of  the  values  thus  found 
in  the  Eqs.  (15)  to  (21)  of  the  preceding  Art.,  which  it  is  not 
necessary  to  reproduce  here. 


Art.  24.— The  Neutral  Curve  for  Special  Cases. 

The  curved  intersection  of  the  neutral  surface  with  a  ver- 
tical plane  passing  through  the  axis  of  a  loaded,  and  originally 
straight,  beam  may  be  called  the  "  neutral  curve."  The  neu- 
tral curve  is  the  locus  of  the  extremities  of  the  ordinates  w  of 
Art.  19 ;  it  therefore  gives  the  deflection  at  any  point  of  the 
beam. 

The  method  of  finding  the  neutral  curve  for  any  particular 
case  of  beam  or  loading  can  be  well  illustrated  by  the  opera- 
tions in  the  following  three  cases. 


172'  ^    V  '    THEORY  OF  FLEXURE.  [Art.  24. 

Case  /. 

This   case   is   s,hown   in   the   accompanying   figure,  which 

represents  a  cantilever  carry- 
ing a  uniform  load  with  a  sin- 
gle weight  IV  at  its  free  end. 
JB  As  usual,  the  intensity  of  the 
uniform  loading  will  be  repre- 
sented 


Fig.1  Measuring  x  and  w  from 

B,  as  shown,  the  general  value 
of  the  bending  moment  is  : 


Integrating  between  x  and  /,  remembering  that 

dw 

for  x  =  /  : 


Hence  : 


The  greatest  deflection,  wu  occurs  for  x  —  /.     Hence  : 

.....  (4) 


UNIVERSITY] 

Art.  24.]  THE  NEUTRAL    CURVE. 


The  greatest  moment,  J/x,  exists  at  ^4,  and  its  value  is  7 


These  equations  are  made  applicable  to  a  cantilever  with 
a  uniform  load  by  simply  making  W=o.  They  then  be- 
come : 


Again,  for  a  cantilever  with  a  single  weight  only  at  its  free 
end,/  is  to  be  made  equal  to  zero  in  the  first  set  of  equations. 
Those  equations  then  become  : 


dx 


174 


THEORY  OF  FLEXURE. 


[Art.  24. 


w_ 

7 

Wl* 

M*=Wl (15) 

The  general  expressions  for  the  shear  and  the  intensity  of 
loading  are  : 

c 

r  r  d^V 

EI-^-^p (17) 

Case  II. 

This  case,  shown  in  the  figure,  is  that  of  a  non-continuous 
beam,  supported  at  each  end,  and  carrying  both  a  uniform  load 

R 

I 
I 

x H 

I 

Air=  B  =\ 


W 


Fig.2 


(whose  intensity  is  p]  and  a  single  weight    W  at  its  middle 
point.     The  reaction  R,  at  either  end,  will  then  be  : 


Art.  24.]  THE  NEUTRAL   CURVE. 


175 


2 

The  general  value  of  the  moment  will  then  be  : 


The  origin  of  x  and  w  is  taken  at  A. 
Remembering  that  : 

dw  I 

—  —  =  o     for    x  =  —  . 
dx  2 


and  integrating  between  the  limits  x  and  —  : 


rrdw       R 

El  -j- 

dx 


R   /  ,      I2  \       p   f         /3  \ 

--{&  ---  —    Z-l^3—   -7T)      •       •       (19) 

2   \          47        6  \          8/ 

Again  integrating  : 


The  greatest  deflection  «/x  occurs  at  the  centre  of  the  span, 
for  which  : 


Hence  : 

34  /  •  •!         .     • 

.      •     (21) 


THEORY  OF  FLEXURE.  [Art.  24. 

The  greatest  moment,  also,  is  found  by  putting  : 


x  =  —  . 

2 


It  has  the  value  : 


These  formulae  are  made  applicable  to  a  non-continuous 
beam  carrying  a  uniform  load  only,  by  putting  W  =  o.  They 
then  become  : 


(23) 


rrdlV  p        X*l  X*  /3  \ 

£/  -r-  .=='<••(  v-  -)....    (24) 

dx  2     \2  12  / 


_  .    5 


384^7  "          8     ' 

^^  ............    (27) 

The  formulae  for  a  beam  of  the  same  kind  carrying  a  single 
weight  at  the  centre,  are  obtained  by  putting/  =  o  in  the  first 


177 


Art.  24.]  THE  NEUTRAL   CURVE. 

set  of  equations.     Those  for  the  greatest  deflection  and  great- 
est moment,  only,  however,  will  be  given.     They  are  : 


Wl* 


(29) 


The  general  values  of  the  shear  and  intensity  of  loading 
are  : 


dM 
J~  =  R-P* (30) 


dx* 


Case  III. 


(30 


The  general  treatment  of  continuous  beams  requires  the 
use  of  the  theorem  of  three  moments.     The  particular  case  to 


a 

B 

3= 


f  ' 

I  c 

- 


F«'g.  3 


be  treated  is  shown  in  Fig.  3.     The  beam  covers  the  three 
spans,  DA,  AB  and  BC,  and  is  continuous  over  the  two  points 
of  support  A  and  B* 
12 


THEOR  Y  OF  FLEXURE.  [Art.  24. 


Let  DA  =  /x 
"    AB  =  /2 


BC  =  /3  j 


h  Let  /2  =  «/f  =  n'L. 


Let  the  intensity  of  the  uniform  load  on  AB  be  represented 
by/  and  let  the  two  single  forces  P  and  P'  only,  act  in  the 
spans  DA  and  BC  respectively.  Also  let  the  two  distances  : 


DE  =  zt  —  a^     and     CF  =  a7 


be  given.  It  is  required  to  find  the  magnitudes  of  the  forces  P 
and  P',  if  the  beam  is  horizontal  at  A  and  B. 

Since  the  beam  is  horizontal  at  A  and  B,  the  bending  mo- 
ments over  those  two  points  of  support  will  be  equal  to  each 
other,  for  the  load  on  AB  is  both  uniform  and  symmetrical. 
Let  this  bending  moment,  common  to  A  and  B,  be  represented 
by  My  As  the  ends  of  the  beam  simply  rest  at  D  and  C,  the 
moments  at  those  two  points  reduce  to  zero. 

Because  the  four  points  D,  A,  B  and  C  are  in  the  same 
level,  the  first  member  of  Eq.  (14),  of  Art.  23,  becomes  equal 
to  zero. 

If  that  equation  be  applied  to  the  three  points  D,  A  and  B, 
fhe  conditions  of  the  present  problem  produce  the  following 
results  : 

Ma  =  o,    Mb  =  Mc  =  M2 
and 


Hence  the  equation  itself  will  become  : 


Art.  24.]  THE  NEUTRAL   CURVE.  179 


&k  +  34)  +  -       (I?  ~  *,°K  +  /  ••  =  O    .     (32) 

Tl  4 

«•  .     _  4P(V  -  »&,  +  tVf  . 

34) 

....     (33) 


/.     Reaction  at     D  =  R,  =  P±±  +      *     ....     (34) 


As  the  origin  of  #,  is  at  /?,  ^r  will  be  measured  from  the 
same  point. 

Separate  expressions  for  moments  must  be  obtained  for  the 
two  portions,  DE  and  EA,  of  /x,  because  the  law  of  loading  in 
that  span  is  not  continuous. 

Taking  moments  about  any  point  of  EA  : 


P  (*-**)  •    •    •    •    •    (35) 

Remembering  that  : 

^  =  o 
dx 

for  x  =  /„  and  integrating  between  the  limits  x  and  /,  : 

El        L    =  S  -  /,«  f~   /:0  +  P*,(*  "   «     (36) 


Again,  remembering  that  w  =  o  for  x  =  /„  and  integrating 
between  the  limits  x  and  /,  : 


l8o  THEORY  OF  FLEXURE.  [Art.  24. 


^  (a*  2#\          P    /*3  2/Is 

EIw  —  —  L  (  ---  /x2^  +  -    -    ---  (  --  /^  H  -- 
2  /         2  3 

(37) 


Taking  moments  about  any  point  in  DE  : 


,       .......     (38) 


(39) 


Making  x  —  ^  in  Eqs.  (36)  and  (39),  then  subtracting  : 


/""  I      /2  /"       2 

0  =  "    2    '  ~ "  T  ^' 


T  (^  "  /l2)  ~  T  fe2  "  /j2)  +  P^lfe  "  /l}  (40) 


Remembering  that  w  —  o  for  ^r  =  o,  and  integrating  be- 
tween the  limits  x  and  o  : 


(,„•  -  /,>  +  /%,  (^  -  /,y.   (41) 

Making  x  =  z,  in  Eqs.  (37)  and  (41),  then  subtracting  : 

-  ,,.)  +         -  (/,.  -  ^  =  O     .      .       (42) 


Art.  24.]  THE  NEUTRAL   CURVE.  l8l 

Putting  the  value  of  Ma  from  Eq.  (33)  in  Eq.  (11),  then  in- 
serting the  value  of  £„  thus  obtained,  in  Eq.  (42),  after 
making  2l  =  a!t  : 


4(2  +  3»)  " 

• 
* 


This  is  the  desired  value  of  P,  which  will  cause  the  beam  to 
be  horizontal  over  the  two  points  of  support  A  and  B  when 
the  span  AB  carries  a  uniform  load  of  the  intensity/. 

By  the  aid  of  Eq.  (43),  Eq.  (33)  now  gives  : 


*  1    12(2  +  3«)  12  12 

It   is  to  be  noticed  that  M2  is  entirely  independent  of  / 
or  ly     Eq.  (43)  also  gives  : 

P  =  Pf^^1 (45) 

Hence  : 

r>/ 

(46) 


Thus  any  of  the  preceding  equations  may  be  expressed  in 
terms  of/  or  P. 


1  82  THEORY  OF  FLEXURE.  [Art.  24. 

RV  also,  becomes  : 

(     } 


_ 
I  -  6a(i  +  a)          12 


or  : 


X,  =  P(l  -  a)  [i  -  I  a(i  +  «)]    .     .     .     (48) 

It  is  clear  that  there  cannot  be  a  point  of  no  bending  in 
DE.  Hence,  the  point  of  contra-flexure  must  lie  between  E 
and  A,  Fig.  3.  In  order  to  locate  this  point,  according  to  the 
principles  already  established,  the  second  member  of  Eq.  (35) 
must  be  put  equal  to  zero.  Doing  so  and  solving  for  x : 

*=     ,,P  a    *>  •     •     (49) 


Since  P  is  always  greater  than  R»  there  will  always  be  a 
point  of  contra-flexure. 

All  these  equations  will  be  made  applicable  to  the  span  BC, 
by  simply  writing  a'  for  a,  /3  for  /„  and  n'  for  n. 

As  an  example,  let  : 

a  =  —     and     n  =  I. 


Eqs.  (43),  (44)  and  (47)  then  give  : 


M  -  -         -  - 
M'~ 


- 

T  16    ' 


Art.  24.]  THE  NEUTRAL    CURVE.  183. 


after  writing  : 

/,  =  12  =  /3  =  /. 

In  general,  the  span  /x  is  called  "  a  beam  fixed  at  one  end, 
simply  supported  at  the  other  and  loaded  at  any  point  with 
the  single  weight,  P." 

Let  it,  again,  be  required  to  find  an  intensity,  *'/',"  of  a  uni- 
form- load,  resting  on  the  span  l»  which  will  cause  the  beam  to  be 
horizontal  at  the  points  A  and  B. 

Since  the  load  is  continuous,  only  one  set  of  equations  will 
be  required  for  the  span.  The  equation  of  moments  will  be  : 


Integrating  between  the  limits  x  and  !I  : 
Integrating  between  the  limits  x  and  o  : 
But,  also,  w  —  o  when  x  —  /x.  Hence  : 


1 84  THEORY  OF  FLEXURE.  [Art.  24. 

This  equation  gives  the  value  Rt  when  p'  is  known.    Making 
x  —  /!  in  Eq.  (50)  and  using  the  value  of  Rt  from  Eq.  (53)  : 


Adapting  Eq.  (32)  to  the  present  case  : 


(ft*  +  plf)  =  o. 
4 


_      __  .  ,          V 

4(2  +  3«) 
Equating  these  two  values  of  M2  : 

/   =    2-P**       .......       (56) 

Thus  is  found  the  desired  value  of  /'.  In  this  case  the 
span  /,  is  called  "  a  beam  fixed  at  one  end,  simply  supported 
at  the  other  and  uniformly  loaded." 

The  points  of  contra-flexure  are  found  by  putting  the 
second  member  of  Eq.  (50)  equal  to  zero  and  solving  for  x, 
after  introducing  the  value  of  R^  from  Eq.  (47).  Hence  : 


or  : 


x  =  o     and     x  =  -/,. 
4 

Between  the  simply  supported   end  and  point   of  contra- 


Art.  24.]  THE  NEUTRAL   CURVE.  185 

flexure  the  beam  is  evidently  convex  downward,  and  convex 
upward  in  the  other  portion  of  the  spans  /t  and  /3,  whether  the 
load  is  single  or  continuous.  Moments  of  different  signs  will, 
then,  be  found  in  these  two  portions,  and  there  will  be  a  maxi- 
mum for  each  sign.  The  location  of  the  sections  in  which 
these  greatest  moments  act  may  be  made  in  the  ordinary  man- 
ner by  the  use  of  the  differential  calculus  ;  but  the  negative 
maximum  is  evidently  M2J  given  by  Eqs.  (44)  and  (55).  On  the 
other  hand  the  positive  maximum  is  clearly  found  at  the  point 
of  application  of  P  in  the  case  of  a  single  load,  and  at  the 
point 


in  the  case  of  a  continuous  load.  These  conclusions  will  at 
once  be  evident  if  it  be  remembered  that  the  portion  of  the 
beam  between  the  supported  end  and  point  of  contra-flexure 
is,  in  reality,  a  beam  simply  supported  at  each  end.  These  mo- 
ments will  have  the  values  : 


In  case  of  a  single  load   if  P  is  given,  and  not  /,  Eq.  (45) 
shows  : 


M,  =  PI,  (i  -  a)a  [i  -  •  i  a(i  +  *)]  . 
The  points  of  greatest  deflection  are  found  by  putting  the 


1 86  THEORY  OF  FLEXURE.  [Art.  24. 

second  members  of  Eqs.  (36),  (40)  and  (48)  each  equal  to  zero, 
and  then  solving  for  x.  They  are  not  points  of  great  impor- 
tance, and  the  solutions  will  not  be  made. 

The  following  are  the  general  values  of  the  shears  for  a 
single  load  on  /,  : 


In  AE-    S  =  El          ==  *,  -  />;  [from  Eq.  (35)]. 


In  ED  ;     S/=  El  ~  -  ^  ;  [from  Eq.  (38)]. 

The  shear  in  /x  for  the  uniform  load/'  is  : 

5'  =  El  ~  =  R,  -  p'X;  [from  Eq.  (50)]. 

Also: 


Intensity  of  load  —  El  -^  —  =  —  /'. 


As  has  already  been  observed,  all  the  equations  relating  to 
the  span  /t  may  be  made  applicable  to  the  span  /3  by  changing 
a  to  a  and  n  to  n'. 

The  span  12  remains  to  be  considered. 

Since  the  bending  moments  at  A  and  B  are  equal  to  each 
other,  and  since  the  loading  is  uniformly  continuous,  half  of  it 
(the  load//2)  will  be  supported  at  A  and  the  other  half  at  B. 
In  other  words,  the  vertical  shear  at  an  indefinitely  short  dis- 
tance to  the  right  of  A,  also  to  the  left  of  B,  will  be  equal  to 

-^-.     Let  x  be  measured  to  the  right  and  from  A.    The  bend- 
ing moment  at  any  section  x  will  be  : 


Art.  24.]  THE  NEUTRAL   CURVE.  l8/ 


£1  =  &,+•*  -. 

dx?  22 


or  : 


-*•)  .    .    .    .    (59) 

t«-t-  & 

Integrating  between  the  limits  x  and  o  : 

dw  p  ( lyX2         x"*  \ 

dx  2  \   2  3  /  ' 

Again  integrating  between  the  same  limits  : 


Since  : 

-4^-  =  o 


for  4  Eq.  (60)  will  give  Ma  independently  of  preceding  equa 
tions.     Following  this  method,  therefore  : 


12 


This  is  the  same  value  which  has  already  been  obtained. 
Introducing  the  value  of  M2  : 

---  •  •  •  (62) 


1  88  THEORY  OF  FLEXUER.  [Art.  24. 


-=  _  j 

dx  2     V   2  3  6 


r-r  Px    f  j  x  4  /^   s 

EIw         =  *-—  (Ijc .     .     .     .     (64) 

12  V  2  2  V   ^; 


The   points   of   contra-flexure    are    found    by  putting   the 
second  member  of  Eq.  (62)  equal  to  zero.     Hence  : 


. 
The  moment  at  the  centre  of  the  span  is  found  by  putting, 

*  =  A 

in  Eq.  (62)  : 

"24" 

This  is  the  greatest  positive  moment. 
The  general  value  of  the  shear  is  : 


C  KTd*W  ^f^  \ 

S  =  El  —f—  =*.(-! x  } 

dx*  \  2  / 

And  the  intensity  of  load  : 


Art.   24.]  THE  NEUTRAL   CURVE.  189 


The  span  /2  is  generally  called  "  a  beam  fixed  at  both  ends 
and  uniformly  loaded." 

It  is  sometimes  convenient  to  consider  a  single  load  at  the 
centre  of  the  span  4  while  the  beam  Remains  horizontal  at  A 
and  B  ;  in  other  words,  to  consider  "  a  beam  fixed  at  each  end 
and  supporting  a  weight  at  the  centre." 

Let  W  represent  this  weight  :  then  a  half  of  it  will  be  the 
shear  at  an  indefinitely  short  distance  to  the  right  of  A  and 
left  of  B.  As  before,  let  x  be  measured  from  A,  and  positive 
to  the  right.  The  moment  at  any  point  will  be  : 

El*£  =  tM-?£-     .....    (65) 
dx^  2 

Integrating  between  x  and  o  : 


(66) 

dx  4 


If*  =  2,  then  will 

2 

dw 

hence  : 

M  -     W/2 

The  general  value  of  the  moment  then  becomes 


19°  THEORY  OF  FLEXURE.  [Art.  25. 

If  x  =  —  in  this  equation,  the  bending  moment  at  the  centre 
(where  W  is  applied)  has  the  value  : 

T/T/7 

Centre  moment  =  — 


8 
> 

Hence,  the  bending  moments  at  the  centre  and  ends  are  each 
equal  to  the  product  of  the  load  by  one  eighth  the  span,  but  have 
opposite  signs. 

A  second  integration  between  x  and  o  gives  : 


I     /  Wl2x2  Wx* 

w  = 


)      .     .     .     .     (68) 


£f\     16  12 

Hence,  the  deflection  at  the  centre  has  the  value  : 

Wl? 


Centre  deflection  = 


192^7  ' 


By  placing  M  =  o,  the  points  of  contra-flexure  are  found  at 
the  distance  from  each  end  : 


r 


Art.  25. — The  Flexure  of  Long  Columns. 

A  "  long  column  "  is  a  piece  of  material  whose  length  is  a 
number  of  times  its  breadth  or  width,  and  which  is  subjected 
to  a  compressive  force  exerted  in  the  direction  of  its  length. 
Such  a  piece  of  material  will  not  be  strained,  or  compressed, 
directly  back  into  itself,  but  will  yield  laterally,  as  a  whole, 
thus  causing  flexure.  If  the  length  of  a  long  column  is  many 


Art.  25.] 


LONG   COLUMNS. 


UNIVERSITY 


times  the  width  or  breadth,  the  failure  in  consequence  of  flex- 
ure will  take  place  while  the  pure  compression  is  very  small. 

As  with  beams,  so  with  columns,  the  ends  may  be  "  fixed," 
so  that  the  end  surfaces  do  not  change  their  position  however 
great  the  compression  or  flexure.  Such  a  column  is  frequently, 
perhaps  usually,  said  to  have  "  flat  "  ends.  If  the  ends  of  the 
column  are  free  to  turn  in  any  direction,  being  simply  sup- 
ported, as  flexure  takes  place,  the  column  is  said  to  have 
"  round  "  ends.  It  is  clear  that  if  the  column  has  freedom  in 
one  or  several  directions,  only,  it  will  be  a  "  round  "  end  col- 
umn in  that  one  direction,  or  those  several  directions,  only.  It 
is  also  evident  that  a  column  may  have'  one  end  "  round  "  and 
one  end  "flat"  or  "fixed." 

In  Fig.  i  let  there  be  represented  a  column  with  flat  ends, 
vertical  and  originally  straight.  After  external  pressure  is 
imposed  at  A,  the  column  will  take  a  shape  similar 
to  that  represented.  Consequently  the  load  P,  at 
At  will  act  with  a  lever  arm  at  any  section  equal 
to  the  deflection  of  that  section  from  its  original 
position.  Let  y  be  the  general  value  of  that  de- 
flection, and  at  B  let  y=yr  Let  x  be  measured 
from  A,  as  an  origin,  along  the  original  axis  of 
the  column.  In  accordance  with  principles  already 
established,  the  condition  of  fixedness  at  each  of 
the  ends  A  and  C  is  secured  by  the  application  of 
a  negative  moment  —  M.  Now  it  is  known  from 
the  general  condition  of  the  column  that  the  curve 
of  its  axis  will  be  convex  toward  the  axis  of  x  at 
and  near  A,  while  it  will  be  concave  at  and  near  B 
(the  middle  point  of  the  column).  Hence,  since  y 
is  positive  toward  the  left,  and  since  the  ordinate  and  its  second 
derivative  must  have  the  same  sign  when  the  curve  is  convex 
toward  the  axis  of  the  abscissas,  the  general  equation  of  mo- 
ments must  be  written  as  follows  : 


Fig,1 


I92 


THEORY  OF  FLEXURE.  [Art.  25. 


Multiplying  by  —  2dy  : 


.-.    El 


c  =  o  because  the  column  has  flat  ends,  and, 


dx 


when  y  —  o.     Also  : 


(0 


(2) 


when  y  = 


Eq.  (2)  now  becomes  : 
rEI 


x  = 


-^y 


(3) 


•     (4) 


Art.  25.]  LONG  COLUMNS. 


193 


(0 


In  this  equation  /  is  the  length  of  the  column.     From  Eq. 
(5)  there  may  be  deduced  : 


(6) 


It  is  to  be  observed  that  Pis  wholly  independent  of  the  de- 
flection, i.  e.,  it  remains  the  same,  whatever  may  be  the  amount 
of  deflection,  after  the  column  begins  to  bend.  Consequently, 
if  the  elasticity  of  the  material  were  perfect,  the  weight  P 
would  hold  the  column  in  any  position  in  which  it  might  be 
placed,  after  bending  begins. 

Eq.  (6)  forms  the  basis  of  "  Hodgkinson's  Formula  "  for 
the  resistance  of  long  columns,  of  which  more  will  be  given 
hereafter.  It  was  first  established  by  Euler. 

Some  very  important  results  flow  from  the  consideration  of 
Fig.  i  in  connection  with  the  preceding  equations. 

The  bending  moment  at  the  centre,  B,  of  the  column  is  ob- 
tained by  placing  y  =  yl  in  Eq.  htf;  its  value  is,  consequently: 


(7) 


Hence  the  bending  at  the  centre  of  the  column  is  exactly  the 
same  (but  of  opposite  sign)  as  that  at  either  end.  Between  A 
and  B,  then,  there  must  be  a  point  of  contra-flexure. 

Putting  the  second  member  of  Eq.  (i)  equal  to  zero,  and 
introducing  the  value  of  M  from  Eq.  (3)  : 


194  THEORY  OF  FLEXURE.  [Art.  25. 

Introducing  this  value  of  y  in  Eq.  (4),  and  bearing  in  mind 
Eq.  (5)  : 


The  points  of  contra-flexure,  then,  are  at  H  and  D,  —  I  and 

£/  from  A 
4 

Hence,  //^  middle  half  of  the  column  (HD)  is  actually  a 
column  with  round  ends,  and  it  is  equal  in  resistance  to  a  fixed- 
end  column  of  double  its  length. 

Hence  writing  /'  for  -  and  putting  2/'  for  /in  Eq.  (6)  : 


(9) 


Eq.  (9)  gives  the  value  of  P  for  a  round-end  column. 

Again,  either  the  upper  three  quarters  (AD)  or  the  lower 
three  quarters  (CH)  of  the  column  is  very  nearly  equivalent  to 
a  column  with  one  end  flat  and  one  end  round,  and  its  resist- 
ance is  equal  to  that  of  a  fixed-end  column  whose  length  is  - 

o 
its  own.  Putting,  therefore  : 


'-=!' 


and  introducing  : 


in  Eq.  (6)  : 


Art.  25.]  LONG  COLUMNS. 


195 


do) 


The  last  case  is  not  quite  accurate,  because  the  ends  of  the 
columns  HC  and  AD  are  not  exactly  in  a  vertical  line. 

In  reality,  the  column  under  compression  may  be  composed 
of  any  number  of  such  parts  as  HD,  with  the  portions  HM  and 
CD  at  the  ends,  thus  taking  a  serpentine  shape,  so  far  as  pure 
equilibrium  is  concerned.  In  such  a  condition  the  column 
would  be  subjected  to  considerably  less  bending  than  in  that 
shown  in  the  figure.  In  ordinary  experience,  however,  the 
serpentine  shape  is  impossible,  because  the  slightest  jar  or 
tremor  would  cause  the  column  to  take  the  shape  shown  in 
Fig.  I.  Hence,  the  latter  case  only  has  been  considered. 

If  r  is  the  radius  of  gyration  and  5  the  area  of  normal  sec- 
tion of  the  column,  Eqs.  (6)  and  (9)  will  take  the  forms  : 


Eq.  (10)  will,  of  course,  take  a  corresponding  form. 

p 
These   equations   evidently  become  inapplicable  when  -^- 

approaches  C,  the  ultimate  compressive  resistance  of  the  ma- 
terial in  short  blocks.  The  corresponding  values  of  ( — j  at 
the  limit,  are  : 


for  fixed   and  round  ends  respectively  ;    other  conditions  of 
ends  will  be  included  between  those  two. 


196  THEORY  OF  FLEXURE.  [Art.  26. 

If,  for  wrought  iron  : 

E  —  28,000,000     and     C  —  60,000, 

the  above  values  become  136  and  68,  nearly. 

Euler's  formula,  therefore,  is  strictly  applicable  only  to 
wrought-iron  columns,  with  ends  fixed  or  rounded,  for  which 
/-f-  r  exceeds  136  and  68,  respectively. 

If,  for  cast  iron  : 

E  —  14,000,000     and     C  =  100,000, 
Eqs.  (n)  give  : 

-  ==  74,      and      -  =  37,  nearly. 

Euler's  formula  evidently  becomes  inapplicable  consider- 
ably above  the  limits  indicated,  since  columns  in  which  —  has 

those  values  will  not  nearly  sustain  the  intensity  C. 

The  analytical  basis  of  "  Gordon's  Formula  "  for  the  re- 
sistance of  long  columns  is  so  closely  associated  with  the 
empirical,  that  both  will  be  treated  together,  hereafter. 


Art.  26. — Graphical  Determination  of  the  Resistance  of  a  Beam. 

The  graphical  method  is  well  adapted  to  the  treatment  of 
beams  whose  normal  sections  are  limited  either  wholly  or  in 
part  by  irregular  curves.  In  Fig.  I  is  represented  the  normal 
section  of  such  a  beam,  the  centre  of  gravity  of  the  section 
being  situated  at  C.  The  lines  HL,  AB  and  DF  are  parallel. 
As  is  known  by  the  common  theory  of  flexure,  the  neutral  axis 
will  pass  through  C. 


Art.  26.] 


GRAPHICAL  METHOD. 


197 


Let  aa  be  any  line  on  either  side  of  AB,  then  draw  the  lines 
ad  normal  to  AB,  having  made  MN  and  //^equidistant  from 
ABK  From  the  points  a,  thus  determined,  draw  straight  lines 
to  C.  These  last  lines  will  include  intercepts,  bb,  on  the  orig- 
inal lines  aa.  Let  every  linear  element  parallel  toA£,  on  each 


side  of  C,  be  similarly  treated.    All  the  intercepts  found  in  this 
manner  will  compose  the  shaded  figure. 

This  operation,  in  reality,  and  only,  determines  an  amount 
of  stress  with  a  uniform  intensity  identical  with  that  developed 
in  the  layer  of  fibres  farthest  from  the  neutral  axis,  and  equal 
to  the  total  bending  stress  existing  in  the  section  ;  this  latter 
stress,  of  course,  having  a  variable  intensity.  HL  represents 
the  layer  of  fibres  farthest  from  the  neutral  surface,  conse- 
quently MNwas  taken  at  the  same  distance  from  AB.  Any 
other  distance  might  have  been  taken,  but  the  intensity  of  the 
uniform  stress  would  then  have  had  a  value  equal  to  that 
which  exists  at  that  distance  from  the  neutral  axis.  Again,  a 


IQ8  THEORY  OF  FLEXURE.  [Art.  26. 

different  intensity  might  have  been  chosen  for  the  stress  on 
each  side  of  AB.  It  is  most  convenient,  however,  to  use  the 
greatest  intensity  in  the  section  for  the  stress  on  both  sides  of 
the  neutral  axis  ;  this  intensity,  which  is  the  modulus  of  rupt- 
ure by  bending,  will  be  represented,  as  heretofore,  by  K. 

Let  c  and  c  be  the  centres  of  gravity  of  the  two  shaded 
figures.  These  centres  can  readily  and  accurately  be  found  by 
cutting  the  figures  out  of  stiff  manilla  paper  and  then  balanc- 
ing on  a  knife  edge.  Let  s  represent  the  area  of  the  shaded 
surface  below  AB,  and  sr  the  area  of  that  above  AB. 

Because  this  is  a  case  of  pure  bending,  the  stresses  of  ten- 
sion must  be  equal  to  those  of  compression.  Hence  : 

Ks  —  Ks'  ;    or,    s  =  sf (i) 

The  moment  of  the  compression  stresses  about  AB  will 
be: 

Ks  X  c'C. 

The  moment  of  the  tensile  stresses  about  the  same  line 
will  be  : 

Ks  X  cC. 


Consequently  the  resisting  moment  of  the  whole  section 
will  be  : 

M  =  Ks  (c'C  +  cC)  =  Ks  X  cc     .     .     .     ,     (2) 

Thus,  the  total  resisting  moment  is  completely  determined. 
In  some  cases  of  irregular  section  the  method  becomes  ab- 
solutely necessary. 

It  is  to  be  observed  that  the  centre  of  gravity,  c  or  c',  is  at 


Art.  27.]       UNEQUAL  COEFFICIENTS  OF  ELASTICITY.  199 

the  same  normal  distance  from  AB  as  the  centre  of  the  actual 
stress  on  the  same  side  of  AB  with  c  or  c'. 


Art.  27. — The   Common   Theory   of  Flexure   with   Unequal  Values   of 
Coefficients   of  Elasticity. 

In  all  cases  it  has  hitherto  been  assumed  that  the  coeffi- 
cient of  elasticity  for  tension  is  equal  to  the  same  quantity  for 
compression.  In  reality,  this  is  exactly  true  for  probably  no 
material  whatever,  though  the  error,  fortunately,  is  not  serious 
for  the  greater  portion  of.  the  material  used  by  the  engineer. 
By  the  aid  of  the  assumptions  used  in  the  common  theory  of 
flexure,  formulae  involving  this  difference  of  coefficients  may 
be  deduced.  As  these  are  of  little  real  value,  however,  a  few 
general  results,  only,  will  be  obtained. 

Let  E  represent  the  coefficient  of  elasticity  for  tension. 

Let  E'  represent  the  coefficient  of  elasticity  for  compres- 
sion. 

As  has  before  been  assumed,  the  normal  sections  of  the 
beam,  which  are  plane  before  flexure,  will  be  taken  as  plane 
and  normal  to  the  neutral  surface  after  flexure.  Also,  as  be- 
fore (Art.  1 8),  let  u  represent  the  rate  of  strain  (strain  for  unit 
of  length  of  fibre)  at  unit's  distance  from  the  neutral  surface ; 
let  the  variable  width  of  the  section  be  represented  by  b,  while 
y  represents  the  variable  normal  distance  of  the  element  bdy 
from  the  neutral  axis  of  the  section.  The  element  of  the  ten- 
sile stress  in  the  section  will  be  : 

Eny  .  b  dy. 

The  elementary  moment  of  the  same  will  be  : 
Euy*  b  dy. 


200  THEORY  OF  FLEXURE.  [Art.  2/. 

In  precisely  the  same  manner,  the  elementary  compressive 
moment  will  be  : 

E  'iifb  dy. 

Consequently,   the  total  resisting  moment   will  have   the 
value  : 


M  =  Ti^E^1  y2b  dy  +  E'{°     y*b  dy\  . 


y 


The  ordinates  y  and  y  are  those  belonging  to  the  extreme 
fibres  of  the  section,  while  K  and  K!  represent  stress  intensi- 
ties in  those  fibres.  The  general  value  of  y  is  also  affected 
with  the  negative  sign  on  the  compression  side  of  the  beam. 

It  has  been  shown  in  Art.  18  that  : 


p 

also,  in  the  case  of  straight  beams,  that  : 

i         d*w 


w  being  the  deflection  and  x  the  abscissa  measured  along  the 
axis  of  the  beam.  For  the  sake  of  brevity,  let  the  quantity  in 
the  brackets  in  the  second  member  of  Eq.  (i)  be  represented 
by  EJ,  in  which,  consequently,  E'  will  be  displaced  by  nE,  n 
being  the  ratio  between  E  and  E'.  Eq.  (i)  may  then  take  the 
form  : 


Art.  27.]        UNEQUAL  COEFFICIENTS  OF  ELASTICITY.  2OI 


or  : 


If  J/ and /are  expressed  in  terms  of  ;r,  w  may  at  once  be 
found.  If,  as  is  usual,  the  section  is  uniform,  then  will  /  be 
constant  and  M,  only,  will  be  a  function  of  x. 

If  the  section  is  rectangular,  b  will  be  constant  and/  will 
take  the  following  value  : 


Because  the  internal  tensile  stress  in  any  section  must  equal 
the  internal  compressive  stress  in  the  same  section  : 

Jyi                        fy' 
by  dy  —  E'u\   by  dy (5) 
o                                         Jo 

Eq.  (5)  will  enable  the  neutral  axis  of  any  section  to  be 
located.  If  the  section  is  symmetrical,  the  neutral  axis  will 
evidently  be  situated  on  that  side  of  the  centre  of  gravity  of 
the  section  on  which  is  found  the  greatest  coefficient  of  elas- 
ticity. 


202  THEORY  OF  FLEXURE.  [Art.  28. 

Art.  28.— Greatest  Stresses  at  any  Point  in  a  Beam. 

If  the  approximate  conditions  on  which  are  based  the 
formulae  found  in  the  latter  part  of  Art.  17  are  assumed,  some 
interesting  and  important  results  may  be  very  easily  obtained. 

The  Eqs.  (13),  (14)  and  (15)  of  Art.  6  are  those  which  lead 
to  the  ellipsoid  of  stress,  and  hence  to  all  of  its  special  cases 
and  consequences.  The  equation  representing  the  ellipsoid  of 
stress  might  first  be  found,  and  then  the  special  form  relating 
to  the  case  considered.  It  will  be  more  simple  and  direct, 
however,  to  use  those  equations  immediately. 

If,  as  in  Art.  17,  a  rectangular  beam  carrying  a  load  at  its 
end  be  assumed,  in  which  : 

Tl=  T3  =  N2  =  N3  =  o, 
Eqs.  (13),  (14)  and  (15)  of  Art.  6  reduce  to  : 

N^  cos  p  -f-  T2  cos  r  =  P  cos  n  ; 
T2  cos  p  =  P  cos  p. 

But  since  all  stress  is  assumed  to  be  found  in  planes-  paral- 
lel to  ZX : 

cos  r  —  sin  p,     and     cos  p  =  sin  n. 
Hence  : 

NI  cos  p  +  Ta  sin  p  —  P  cos  n (i) 

T2  cos  p  =  P  sin  7t (2) 

in  which  P  is  the  intensity  of  the  resultant  stress  on  any  plane 


Art.  28.]  GREA  TEST  STRESS  A  T  ANY  POINT.  2O3 

at  any  point;  /  the  angle 'which  the  normal  to  that  plane 
makes  with  the  axis  of  X  (the  axis  of  the  beam) ;  and  TT  the 
angle  which  the  direction  of  P  makes  with  the  same  axis. 

Let  it  first  be  required  to  find  the  plane,  at  any  point,  on 
which  the  normal  or  direct  stress  is  the  greatest. 

It  is  known  from  the  theory  of  internal  stress  that  this 
greatest  normal  stress  will  be  the  resultant  and,  hence,  a  prin- 
cipal stress.  Hence  the  relation  :  n  =  p  ;  or: 

N,+  T2tanp  =  P (3) 

T,  =Ptanp (4) 

If  F  is  the  weight  carried  by  the  beam  at  its  end;  /the 
moment  of  inertia  of  the  beam's  cross  section ;  and  d  its  half 
depth,  or  greatest  value  of  z,  it  has  been  shown  in  Arts.  17 
and  1 8  that  : 

N^^-z,    and     r..-^(*.-*0     •     •     •     (5) 

Inserting  the  value  of  P  from  Eq.  (4)  in  Eq.  (3)  : 
T;  —  Ta  tan2  p  =  Nt  tan  p. 

.-.     tan*p  +  ^  tanp  =  i. 

J-  2 

Solving  this  quadratic  equation  and  then  inserting  the 
values  of  T3  and  Nt  from  Eq.  (5)  : 


tan  p  = -JT—     —  ±  ~/a ;r 

a    —  2  a-    —  *» 


204  THEOR  Y  OF  FLEXURE.  [Art.  28. 

This  value  of  tan  p  put  in  Eq.  (3),  or  Eq.  (4),  will  give  the 
greatest  value  of  the  direct  or  normal  stress  (also  resultant)  at 
any  point  in  the  beam. 

At  tJie  exterior  surface,  d—Z',  hence  : 

tan  p  =  o     or     —  oo. 

Since  for  this  point  T2  =  o,  the  first  value  gives,  by  Eq.  (3), 
P—  Nr  The  second  value,  by  Eq.  (4),  gives,  P=  o.  These 
results  might  have  been  anticipated. 

At  the  neutral  surface,  Z  —  Q\  hence  : 

tan  p  —  ±  i  =  tan  ±  45°. 

Hence,  at  the  neutral  surf  ace  there  are  two  planes  on  which  the 
stress  is  wholly  normal,  and  these  planes  make  angles  0/"  45  °  with 
the  neutral  surface,  or  90°  with  each  other  (i.e.,  they  are  prin- 
cipal planes). 

Since  N^  =  o  at  the  neutral  surface,  either  of  the  Eqs.  (3) 
or  (4)  gives  : 


(7) 


Hence  each  of  these  normal  or  principal  stresses  equals  in  inten- 
sity that  of  the  transverse  or  longitudinal  shear  at  the  neutral 
surface  ;  also,  one  of  these  principal  stresses  is  a  tension  and  the 
other  a  compression. 


is  the  equation  of  the  locus  of  the  point  of  constant  greatest 


Art.  28.]  GREATEST  STRESS  AT  ANY  POINT. 


205 


normal  intensity  of  stress,  if  P  be  taken  constant  and  equal  to 
any  possible  value. 

Let  it  next  be  required  to  find  the  plane  of  greatest  shear  at 
any  point  in  the  beam,  and  the  value  of  that  shear. 

The  shear  on  any  plane  will  be  : 


P  sin  (n  —  p)  =  T 


(8) 


Multiplying  Eq.  (i)  by  (—  sin  p)  and  Eq.  (2)  by  cos p,  then 
adding  : 

—  A",  cos  p  sin  p  -f  T2  (cos2  p  —  sin*  p}  =  P(sin  n  cos  p 
—  cos  7t  sin  p)  =  P  sin  (n  —  /)  —  T. 


.-.     T  =  -        sin  2p  +  T2  cos  2p     .      .     .     .     (9) 


It  is   now  required   to   find  what  value  of  /  will  make  the 
general  value  of  T  [given  by  Eq.  (9)]  a  maximum.     Hence  : 


—  -  —  —  —  NT.  cos  2p  —  2  T2  sin  2p  —  o. 
dp 


ta;/  2/> 

/.      COS   2p 


*=-  *  =- 


** 


V*2;2  4- 


~  r 


.      .      .      (10) 


Eqs.  (10)  give  the  value  of/  which  is  to  be  placed  in  Eq. 
(9),  in  order  to  obtain  the  greatest  value  of  T  at  any  point  of 
the  beam. 


2O6  THEORY  OF  FLEXURE.  [Art.  28. 

From  Eq.  (9)  : 


T  =  Ta  cos  2p  |  -  ~±-  tan  2p  +  1  1 


•'•        T  =  ±          V*2*8  +  (d2  -  zj     ....     (II) 

At  the  exterior  surfaces  of  the  beam  : 

z  =  ±  d, 
Hence  : 

T  Fxd  AT, 

r=±  ir  =  ±  T" 

For  this  case,  also  : 

cos  2p  =  o,     or    /  =  45°. 

Hence,  #/  /^  exterior  surfaces  of  the  beam  the  planes  of 
greatest  shear  make  angles  of  45°  with  the  axis  of  the  beam, 
and  the  intensity  of  the  shear  is  half  that  of  the  direct  stress  at 
the  same  place. 

At  the  neutral  surface  :  z  =  o.     Hence  : 


Fd* 

T  ==  ±  — j  =  T3 ;    and  cos  2p  =  ±  i. 

Hence,  2p  —  o  or  180°  ;  or/  =  o  or  90°  ;  z>.,  the  planes  of 
greatest  shear  are  the  transverse  and  longitudinal  planes,  and 
the  greatest  shear  itself  is,  consequently,  the  transverse  or  lon- 
gitudinal shear. 


Art.  28.]  GREA TEST  STRESS  AT  ANY  POINT.  2O/ 

If  T  is  given  any  possible  value  and  considered  constant, 
Eq.  (i  i)  will  give  the  locus  of  the  point  of  constant  greatest 
shear. 

The  result  expressed  in  Eq.  (7)  is  of  great  value  in  deter- 
mining the  thickness  of  the  web  of  flanged  beams,  as  will  be 
seen  hereafter. 


PART  II.— TECHNICAL. 


CHAPTER  V. 

TENSION. 


Art.  29. — General   Observations. — Limit  of  Elasticity. 

HITHERTO,  certain  conditions  affecting  the  nature  of  elastic 
bodies  and  the  mode  of  applying  external  forces  to  them,  have 
been  assumed  as  the  basis  of  mathematical  operations,  and 
from  these  last  have  been  deduced  the  formulae  to  be  adapted 
to  the  use  of  the  engineer.  These  conditions,  in  nature,  are 
never  realized,  but  they  are  approached  so  closely,  that,  by  the 
introduction  of  empirical  quantities,  the  formulae  give  results 
of  sufficient  accuracy  for  all  engineering  purposes  ;  at  any  rate, 
they  are  the  only  ones  available  in  the  study  of  the  resistance 
of  materials. 

In  determining  the  quantity  called  the  "  coefficient  of  elas- 
ticity," it  is  supposed  that  the  body  is  perfectly  elastic,  i.e.9  that 
it  will  return  to  its  original  form  and  volume  when  relieved  of 
the  action  of  external  forces,  also,  that  this  "  coefficient  "  is 
constant.  There  is  reason  to  believe  that  no  body  known  to 
the  engineer  is  either  perfectly  elastic  or,  possesses  a  perfectly 
constant  coefficient  of  elasticity.  Yet,  within  certain  not  well 
defined  limits  the  deviations  from  these  assumptions  are  not 
sufficiently  great  to  vitiate  their  great  practical  usefulness. 


Art.  29.]  LIMIT  OF  ELASTICITY. 


209 


The  "  not  well  defined  "  limit  for  any  one  given  material  is 
called  its  "  limit  of  elasticity,"  or  "  elastic  limit."  The  "  limit 
of  elasticity,"  then,  may  be  defined  as  that  degree  of  stress 
within  which  the^  coefficient  of  elasticity  is  essentially  constant 
and  equal  to  the  ^stress  divided  by  the^strain.  /^v^Jdo 

In  some  materials,  like  many  grades  of  wrought  iron  and 
steel,  the  limit  of  elasticity  approximates,  to  a  greater  or  less 
degree,  to  the  condition  of  a  well  defined  point.  If  a  piece  of 
such  a  material  is  subjected  to  stress  in  a  testing  machine,  at 
the  elastic  limit,  the  amount  of  strain  caused  by  a  given  incre- 
ment of  stress  will  be  observed,  comparatively  speaking,  to 
rapidly  increase.  This  increase  may  be  uniform  for  a  consider- 
able range  of  stress,  but  it  finally  becomes  irregular,  after 
which  failure  takes  place. 

In  other  materials,  there  seems  to  be  no  simple  relation 
between  stress  and  strain  for  any  condition  of  stress  whatever. 
For  such  a  material  it  obviously  is  impossible  to  assign  either 
any  definite  elastic  limit  or  coefficient  of  elasticity. 

Between  these  limits,  of  course,  all  grades  of  material  are 
found. 

It  should  be  stated  that  some  authorities  have  given  arbi- 
trary definitions  of  the  elastic  limit,  and  that  these  definitions 
have  been  very  much  used.  Wertheim  and  others  have  con- 
sidered the  elastic  limit  to  be  that  force  which  produces  a  per- 
manent elongation  of  0.00005  °f  tne  length  of  a  bar.  Again, 
Styffe  defines,  as  the  limit  of  elasticity,  a  much  more  compli- 
cated quantity.  He  considers  the  external  load  to  be  gradu- 
ally increased  by  increments,  which  may  be  constant,  and  that 
each  load,  thus  attained,  is  allowed  to  act  during  a  number  of 
minutes  given  by  taking  100  times  the  quotient  of  the  incre- 
ment divided  by  the  load.  Then  the  "  limit  of  elasticity  "  is 
"  that  load  by  which,  when  it  has  been  operating  by  successive 
small  increments  as  above  described,  there  is  produced  an  in- 
crease in  the  permanent  elongation  which  bears  a  ratio  to  the 
length  of  the  bar  equal  to  o.oi  (or  approximates  most  nearly 


210  TENSION.  [Art.  30. 

to  o.oi)  of  the  ratio  which  the  increment  of  weight  bears  tc» 
the  total  load."  (Iron  and  Steel,  p.  30.)  f^rif^  y.0"1^  1*1 

The  most  natural  value,  however,  seems  to  re  that  stress 
which  exists  at  the  point  where  the  ratio  between  stress  and 
strain  ceases  to  be  essentially  constant,  though  the  assignment 
of  the  precise  point  be  difficult  in  many  cases  and  impossible 
.in  some  ;  and  in  that  sense  it  is  here  used,  though  seldom  in 
ordinary  testing. 

Again,  in  the  common  theory  of  flexure,  modes  of  appli- 
cation of  external  forces  and  a  constitution  of  material  are 
assumed,  which  are  never  realized  ;  yet  the  resulting  formulae 
are  of  inestimable  value  to  the  engineer. 

Finally,  it  will  be  shown  in  the  first  section  of  Art.  32  that 
it  is  in  general  impossible  to  produce  a  uniform  intensity  of 
stress  in  a  normal  cross  section  of  a  body  subjected  to  pure 
tension,  and,  consequently,  that  the  ultimate  resistance,  as 
experimentally  determined,  is  a  mean  intensity  which  may  be, 
and  usually  is,  considerably  less  than  the  maximum  sustained 
by  the  test  piece. 

These  general  observations  are  to  be  carefully  borne  in 
mind  in  connection  with  all  that  follows. 


Art.  30. — Ultimate   Resistance. 

After  a  piece  of  material,  subjected  to  stress,  has  passed  its 
elastic  limit,  the  strains  increase  until  failure  takes  place.  If 
the  piece  is  subjected  to  tensile  stress,  there  will  be  some  de- 
gree of  strain,  either  at  the  instant  of  rupture  or  somewhat 
before,  accompanied  by  an  intensity  of  stress  greater  than  that 
existing  in  the  piece  in  any  other  condition.  This  greatest  in- 
tensity of  interna,!  resistance  is  called  the  "  Ultimate  Resist- 
ance." c  $£~~7^ 

In  very  ouctile  materials  this  point  of  greatest  resistance  is 
found  considerably  before  rupture  ;  the  strains  beyond  it  in- 


Art.  31.]  DUCTILITY  AND   SET.  211 

creasing  very  rapidly  while  the  resistance  decreases  until  sepa- 
ration takes  place. 

The  ultimate  resistances  of  different  materials  used  in  en- 
gineering constructions  can  only  be  determined  by  actual  tests, 
and  have  been  the  objects  of  many  experiments. 

It  has  been  observed  in  these  experiments  that  many  in- 
fluences affect  the  ultimate  resistance  of  any  given  material, 
such  as  mode  of  manufacture,  condition  (annealed  or  unan- 
nealed,  etc.),  size  of  normal  cross  section,  form  of  normal  cross 
section,  relative  dimensions  of  test  piece,  shape  of  test  piece, 
etc.  In  making  new  experiments  or  drawing  deductions  from 
those  already  made,  these  and  similar  circumstances  should  all 
be  carefully  considered. 


Art.  3i.— Ductility.— Permanent   Set. 

One  of  the  most  important  and  valuable  characteristics  of 
any  solid  material  is  its  "  ductility,"  or  that  property  by  which 
it  is  enabled  to  change  its  form,  beyond  the  limit  of  elasticity, 
before  failure  takes  place.  It  is  measured  by  the  permanent 
"  set,"  or  stretch,  in  the  case  of  a  tensile  stress,  which  the  test 
piece  possesses  after  fracture  ;  also,  by  the  decrease  of  cross 
section  which  the  piece  suffers  at  the  place  of  fracture. 

In  general  terms,  i.e.,  for  any  degree  of  strain  at  which  it 
occurs,  "  permanent  set "  is  the  strain  which  remains  in  the 
^iece  when  the  external  forces  cease  their  action.  It  will  be 
seen  hereafter  that  in  many  cases,  and  perhaps  all,  permanent 
set  decreases  during  a  period  of  time  immediately  subsequent 
to  the  removal  of  stress.  Indeed,  in  some  cases  of  small  strains 
it  is  observed  to  disappear  entirely. 

Some  experimenters,  with  the  aid  of  very  delicate  meas- 
uring apparatus,  have  observed  permanent  set  even  within  what 
is  ordinarily  termed  the  limit  of  elasticity,  and  have  been  led 
to  believe  that  a  very  small  permanent  set  exists  with  any  de- 


212  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

gree  of  stress  whatever.  In  such  cases,  however,  it  is  probable 
that  the  greater  part  or  all  of  the  permanent  set  disappears 
after  the  lapse  of  a  few  hours. 


Art.  32. — Wrought   Iron. — Coefficient   of  Elasticity. 

Before  considering  the  experimental  results  which  are  to 
follow,  it  will  be  interesting  as  well  as  important  to  examine 
some  of  the  circumstances  which  attend  the  experimental  de- 
termination of  the  coefficient  of  elasticity. 

If  tensile  stress  is  uniformly  distributed  over  each  end  of 
a  test  piece,  it  will  not  be  so  distributed  over  any  other  normal 
section.  For  since  lateral  contraction  takes  place,  the  exterior 
molecules  of  the  piece  must  move  towards  the  centre.  But  if 
this  motion  takes  place,  the  molecules  in  the  vicinity  of  the 
centre  must  be  drawn  farther  apart,  or  suffer  greater  strains, 
than  those  near  the  surface. 

Hence  the  stress  will  no  longer  be  uniformly  distributed, 
but  the  greatest  intensity  will  exist  at  the  centre  and  the  least 
at  the  surface  of  the  piece.  These  effects  will  evidently  in- 
crease, for  a  given  kind  of  cross  section,  with  its  area.  But  the 
stretch,  or  strain,  from  which  the  coefficient  of  elasticity  is 
computed,  is  measured  on  the  surface  of  the  piece,  and  corre- 
sponds, as  has  just  been  shown,  to  an  intensity  of  stress  less 
than  the  mean,  while  the  latter  is  actually  used  in  the  compu- 
tation. In  the  notation  of  Eq.  (i),  Art.  2,  /  is  too  great  and  / 
too  small ;  hence  E  will  be  too  large. 

As  these  effects  increase  with  the  area  of  the  cross  section, 
while  other  things  are  the  same,  larger  bars  should  give  greater 
coefficients  of  elasticity  than  smaller  ones. 

These  effects  will  evidently  be  intensified,  also,  if  the  ex- 
ternal force  is  applied  with  its  greatest  intensity  near,  or  at, 
the  centre  of  the  bar,  as  is  the  case  in  testing  eye-bars. 

Again,  on  the  other  hand,  if  the  ends  of  the  test  piece  are 


Art.  32.] 


COEFFICIENT  OF  ELASTICITY. 


2I3 


gripped  on  the  surface,  or  skin,  as  is  usually  the  case  with 
small  pieces,  these  effects  will  be  very  much  modified,  and 
possibly  entirely  counteracted,  so  that  the  greatest  intensity 
will  exist  at  the  surface.  In  the  latter  case,  the  resulting  co- 
efficient would  be  too  small. 

Between  these  extreme  cases,  all  grades  will  be  found. 

From  these  considerations,  it  is  clear  that  the  manner  of 
gripping  the  test  piece,  length,  character  and  area  of  cross  sec- 
tion all  affect  the  value  of  the  coefficient  of  elasticity,  and 
should  be  given  in  connection  with  the  latter. 

These  conclusions  apply  to  any  other  material,  as  well  as 
to  wrought  iron. 

Table  I.  gives  the  results  of  some  experiments  made  by  the 
Phoenix  Iron  Co.,  of  Phoenixville,  Penn.,  on  some  flats  and 
rounds  of  the  dimensions  shown  in  the  column  headed  "  Size." 

TABLE  I. 


NO.    OF   BARS. 

SIZE. 

LENGTH. 

STRETCH. 

>^ 

£. 

Inches. 

Ft.     In. 

Inches. 

Pounds. 

Pounds. 

12 

4     x  i-jj 

35     o 

0.2692 

20,000.00 

3I,203,CCO.OO 

9 

4     x  lA 

27     6 

0.2033 

32,464,700.00 

24 

3?  x  ii 

35     o 

0.2500 

33,600,000.00 

24 

3z  x  ii 

35     o 

0.2617 

32,098,000.00 

23 

3     x    f 

35     o 

0.2587 

32,470,000.00 

24 

3     x     4 

35     o 

0.2633 

31,902,000.00 

24 

2X1 

,      24     gi- 

o. 1948 

30,544,000.00 

36 

210          J 

ll     9 

0.0953 

29,380,000.00 

68 

210       / 

II   II 

0.0998 

28.056,000.00 

120 

*l°    J* 

ii     9 

0.0947 

29,567,000.00 

48 

2iofcJ 

ii     9 

0.0955 

29,319,000.00 

72 

afoT 

ii     9 

0.0940 

29,787,000.00 

48 

2^0 

ii     9 

0.1008 

<           i 

27,777,777-00 

The  column   "  p  "  is  the  intensjty^p^rso^iaajreinch  which 
caused  the  stretches  shown  in  the  column  headed  "  Stretch." 
From  Eq.  (i)  of  Art.  2  : 


214  WROUGHT  IRON  IN  TENSION.  [Art.  32. 


(0 


In  this  case,  for  any  individual  bar  : 

Stretch 
~~  Length   'f 

remembering  that  the  stretch  and  length  must  be  reduced  to 
the  same  unit. 

Let  the  above  formulae  be  applied  to  the  twenty-four  bars 
3  X  ^  inches  x  (35  ft.  =  420  ins.)  long. 


,-.       20000  X  420  , 

E  =  -  —  =  31,002,000.00  pounds. 

0.2633 


The  other  values  are  found  in  precisely  the  same  way. 
The  quantities  in  the  column  E  are  the  averages  of  the  num- 
ber of  experiments  given  in  the  extreme  left  hand  column. 
The  fact  that  the  results  are  the  averages  of  a  great  number 
of  experiments  gives  the  table  peculiar  value.  This  table 
is  taken  from  "  Useful  Information  for  Architects  and  En- 
gineers," published  by  the  Phoenix  Iron  Co.  The  following 
reference  to  the  table  is  taken  from  the  same  source  :  "  The 
annexed  table  gives  the  results  attained  in  testing  with  the 
proof  load  of  20,000  pounds  per  square  inch,  a  number  of  bars 
for  the  International  Bridge  over  the  Niagara  River,  near 
Buffalo,  N.  Y.  The  recovery  of  each  bar,  after  the  removal  of 
the  load,  was  perfect,  no  permanent  set  occurring  at  less  than 
25,000  pounds.  It  will  be  observed  that  the  stretch  per  foot  of 
the  flat  bars  is  less  than  that  of  the  rounds,  giving  them  higher 
moduli  of  elasticity."  It  is  interesting  and  important  to  ob- 
serve this  last  point. 


Art.  32.] 


COEFFICIENT  OF  ELASTICITY. 


215 


It  is  to  be  observed,  finally,  that  these  coefficients  of  elas- 
ticity are  determined  for  one  intensity  of  stress,  only,  i.e., 
20,000.00  pounds  per  square  inch.  It  is  probable  that  values  a 
little  different  might  be  given  by  other  intensities. 

Table  II.  contains  coefficients  of  elasticity  for  tension,  in 
pourids^ per  square  inch,  computed  from  data  given  in  the 
"  Report  of  the  Committee  of  the  Franklin  Institute  .  .  .  Part 
II.,  containing  the  Report  of  the  sub-committee  to  whom  was 
referred  the  examination  of  the  strength  of  the  materials  em- 
ployed in  the  construction  of  steam  boilers." 


TABLE  II. 


X 

LENGTHWISE 

• 

MODE   OF 

b 
" 

OR 

L. 

LI. 

/• 

T. 

E. 

MANUFACTURE. 

6 

CROSSWISE. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

V<fr      32 

Cross. 

Rolled 

23.00 

.00883 

'12,768 

33,280,000 

32 

** 

23.00 

.0174 

'  ~?  1  9,  1  52 

25,220,000  v 

32 
32 

„ 

23.00 
23.00 

.0274 
.039 

'^25,536 
$  31,920 

52,200 

21,450,000  /C 
18,832,000    V 

32 

" 

23.00 

.0525 

7  38,304 

16,777,000     ! 

\ 

Length. 

Hammered 

23.00 
24.5 

.0669 
.01354 

y-  44,688 
V  26,600 

: 

15,373.00° 
48,146,000  ' 

48 

** 

24.5 

.02882 

6  39,900 

33.955^000  J— 

48 
48" 

" 

•24.5 
24.5 

.°47 
.072 

H    50,469 
''    52,725 

•59,7™ 

26,294,000  7 
17,927,000'^ 

48 

" 

24.5 

.0958 

T-55,100 

14,106,000  '  t 

«« 

24.5 

.144 

'   57.832 

9,831,400 

27 

Cross. 

Rolled 
Hammered 

24.2 
24.0 

.0297 
•°449 

o  30,840 
//  28,250 

55.636 

25,168,000^ 
15,086,000  'i 

53 

** 

•• 

24.0 

0714 

'»  26,270 

56,062 

20,070,300 

56 

" 

Puddled 

23-7 

.°°59 

~i_  28,000 

58,964 

112,476,000  1 

65 
85 

Length. 



24.25 
22.10 

.0161 
.0127 

?  31,540 
li  25.280 

5L255 

47,499,000  . 
44,161,000  ^ 

This  report  was  published  in  the  "  Journal  of  the  Franklin 
Institute,"  for  1837.  The  data  for  the  computations  were  taken 
from  pages  260  to  275  of  Vols.  XIX.  and  XX.  of  that  journal. 
The  test  specimens  were  about  0.25  square  inch  in  area  of 
normal  section  and  were  cut  from  boiler  plate  across  or  along 
the  fibre,  as  is  indicated  in  the  table.  The  experiments  were 


2l6  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

made  in  1832  and  1833.  About  5  per  cent  was  allowed  for  the 
friction  of  the  machine.  It  is  presumed  from  the  report  (which 
is  not  very  clear  on  some  important  points)  that  the  smallest 
values  of  "/,"  in  the  various  experiments,  indicate  the  elastic 
limit  of  the  different  bars. 

The  column  LI  gives  the  extensions  of  the  bars  whose 
lengths  appear  in  the  column  Z. 

The  column  "/  "  gives  the  intensities  of  stress  per  square 
inch  for  which  the  corresponding  values  of  the  coefficients  of 
elasticity  (in  the  column  JS)  have  been  computed. 

In  computing  E,  there  is  to  be  put  in  Eq.  (i)  : 

LI   c       , 
-=-   for  /. 


Bars  No.  32  and  48  show  the  decrease  of  the  coefficient  of 
elasticity  for  degrees  of  stress  up  to  the  ultimate  resistances 
per  square  inch  of  original  section  shown  in  column  T. 

The  values  of  E  can  scarcely  be  considered  more  than  ap- 
proximate, since  the  manner  of  holding  the  specimen,  taken  in 
connection  with  the  method  of  measuring  Z/,  would  probably 
make  the  elongation  either  somewhat  greater  or  less  than  that 
belonging  to  the  length  L.  This  might  make  some  of  the 
values  of  E  greater  than  they  ought  to  be.  That  belonging  to 
bar  56  apparently  indicates  an  error,  but  it  is  believed  not  to 
be  in  the  computations.  Undoubtedly,  however,  many  of  the 
irons,  at  that  early  day,  were  very  stiff  and  hard. 

Prof.  Woodward,  in  "  The  Saint  Louis  Bridge/'  gives  the 
results  of  67  experiments  on  specimens  varying  from  6  to  18 
inches  long  and  from  0.45  inch  to  1.13  inches  in  diameter,  from 
17  different  producers.  In  these  results  the  range  of  variation 
was  very  great  ;  in  fact  the  coefficient  of  tensile  elasticity  varied 
from  9,500,000  Ibs.  per  sq.  in.  to  65,500,000,  and  some  of  the 
widest  variations  wrere  in  specimens  of  the  same  brand. 

Table  III.  gives  the  results  of  the  experiments  of  Mr.  Eaton 


Art.  32.] 


COEFFICIENT  OF  ELASTICITY. 


217 


TABLE  III. 


Tensile  Experiments  on  two  Annealed  "Best"   Wrought  Iron  Bars 
ten  feet  long  and  one  inch  square. 


BAR   NO.   I. 

BAR  NO.    2. 

P- 

LI. 

Sets. 

E. 

P- 

LI. 

Sets. 

E. 

Inches. 

Inches. 

Inches. 

Inches. 

2,668 

.00986 



32,457,000 

1,262 

.00520 



29,125,000 

5.335 
8,003 

.02227 
.o3407 

.000305 

28,198,000 
28,180,000 

2,524 
3,786 

.01150 
.01690 

.00050 

26,347,000 
26,851,000 

10,670 

.04556 

.000407 

28,101,000 

5,047 

.02240 

.00060 

26,990,000 

13.338 

.05705 

.000509 

28,056,000 

6,309 

.02772 

.00050 

27,312,000 

16,005 

.06854 

.000610 

28,020,000 

7.571 

.03298 

.00045 

27,551,000 

.07993 

.000813 

28,033,000 

8.833 

.03790 

.00050 

27,953,ooo 

21,340 

t.  24,008  » 
26,676  . 

.09193 
.10485 
.12163 

.001525 
.003966 
.009966 

27,855,000 
27,475,000 
26,308,000 

10,095 

JI,357 
12,619 

.04300 
.04854 
•05370 

.00050 
.00070 

28,198,000 
28,077,000 
28,199,000 

29,343 

.15458 

.031424 

22,782,000 

13,880 



27,984,000 

32,011 

.26744 

14,361,000 

15,142 

06480 



28,041,000 



.28271 

.13566 

16,404 

.06980 



28,186,000 

in  5  minutes 

17,666 

•07530 

.00130 

28,153,000 

34,678 
37,346 

Repeated 
40,013 

.5M8 
1.095. 
I  .  1949 

1.220 

in  5  minutes 

.36864 
1.01695 
1.02966 
1.093 

8,083,000 
4,077,000 

3,924,000 

18,928 
20,190 
21,452 
22,713 

7*25,237 

.08170 
.08740 
.09310 
.09920 
.10570 
.11250 

.00270 
.00410 
.00680 

27,794,000 
27.734,ooo 
27,644,000 
27,474,000 
27,213,000 
26,919,000 

Repeated 

1.411 



.^__ 

26,499 

.  i  2040 



26,420,000 

and  left  on 

after  i  hours 

27,761 

.12880 

.0120 

25,872,000 

29,023 

.  I45OO 

• 

23,986,000 

44 

i  .424 

• 



30,285 

.1991 



18,244,000 

after  2  hours 

30,285 

.2007 



18,030,000 

after  5  min. 

1.433 
after  3  hours 

30,285 

.2018 

.0736 

- 

after  10  min. 

L434 
after  4  hours 

30,285 

.2054 

.0774 

___ 

after  15  min. 

1.436 
after  5  hours 

Repeated 

.2080 

.0796 



after  20  min. 

1.437 
after  6  hours 

M 

.2006 

.0814 



after  i  hour 

** 

1.443 

•^—^— 

—  — 

after  7  hours 

" 

.2366 

.1082 



M 

1.443 





after  17  hours 

after  8  hours 

31,546 

.242 

.1083 

15,617,000 

after  5  mm. 

L443 

u 

after  9  hours 

Repeated 

.2449 
after  5  min. 

.mi 



1.443 
after  10  hours 

32,808 

.5506 

.4141 

7,132,000 

42,681 

2.148 
in  5  minutes 

1.983 

2,384,000 

Repeated 

.7024 
after  5  mm. 

.5635 



Repeated 

.    2.339 
in  5  minutes 

" 

.70966 
after  10  mm. 

.6558 



218 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


TABLE    III.— Continued. 


BAR  NO.  I. 

BAR   NO.    2. 

/• 

LL 

Sets. 

E. 

P- 

LL 

Sets. 

E. 

Inches. 

Inches. 

Inches. 

Inches. 

Repeated 

2.383 
in  10  minutes 

2.212 



Repeated 

1.014 
after  13  min. 

.866 



" 

2.428 
after  46  hours 

2.237 



34,070 

1.346 
after  i  min. 



2,839,000 

45,348 

2.580 
after  5  min. 

2.377 

2,I09,"000 

34.070 

i  .400 
after  2  min. 





Repeated 

2.605 
after  i  hour 



— 

34,070 

i.  600 

1.44 



«        . 

2.606 
after  2  hours 



— 

Repeated 

1.65 
after  i  min. 





" 

2.606 
after  19  hours 

2.403 

— 

" 

1.786 
after  i  hour 

1.628 



48,016 

2.975 
after  5  min. 

2.733 

1,936,000 

35,332 

2.04 
after  5  min. 

1.874 

2,078,000 

Repeated 

3.019 
after  i  hour 



— 

Repeated 

2.18 
after  5  min. 

2.OI 

~~ 

M 

3.029 





" 

2.254 

2.08 



after  1  1  hours 

36,594 

2.54 



1,743,000 

50,684 

.    4.195 
in  xo  minutes 

3.941 

1,448,000 

37,856 

after  6  min. 
2.894 

1,571,000 

Repeated 

4.226 



— 

" 

4.227 





in  7  hours 

" 

4-227 





in  12  hours 

53,35i 

Broke 



— 

Hodgkinson  on  the  tensile  elasticity  and  permanent  set  of  two 
wrought  iron  bars.  The  coefficients  of  elasticity  E  have  been 
computed  from  the  data  contained  in  the  first  three  columns  as 
given  by  Mr.  B.  B.  Stoney  in  his  "  Theory  of  Strains  in  Girders 
and  similar  Structures."  The  following  is  the  notation  used  : 

/  =  pounds  per  square  inch  ; 

-A-  LI  =  total  elongation,  or  strain,  for  the  bar  ; 

"  Sets  "  =  permanent  set  ;  £ 

E.  =  coefficient  of  tensile  elasticity  =  /  x  L  -^  LL  - 


Art.  32.]  COEFFICIENT  OF  ELASTICITY.  2ig 

These  experiments  show  some  very  interesting  results. 

In  the  first  place  permanent  sets  were  observed  with  the 
low  intensities  of  stress  of  8,003  anc*  3,786  pounds,  and  it  be- 
comes a  question  whether  permanent  sets  would  not  have  been 
observed  with  lower  intensities  and  more  delicate  apparatus, 
at  least  for  a  short  time  after  the  material  is  subjected  to 
stress. 

In  both  bars  the  largest  value  of  E  is  found  for  the  smallest 
intensity  of  stress.  In  bar  No.  I,  the  values  of  E  decrease, 
with  one  exception,  regularly  from  the  greatest.  In  bar  No. 
2,  however,  greater  irregularity  is  observed  ;  there  are  two 
maxima,  one  for  the  intensity  1,262  pounds,  and  the  other  for 
about  12,000  with  nearly  regular  gradations  from  these  values. 

Considering  the  whole  range  in  both  bars,  E  may  be  con- 
sidered nearly  constant  until  an  intensity  of  about  24,000 
pounds  per  square  inch  is  reached  in  each  case ;  it  then  begins 
to  fall  off  very  rapidly.  24,000  pounds  per  square  inch,  then, 
may  be  considered  about  the  limit  of  elasticity  for  both  bars. 

It  is  very  important  to  observe  the  increase  of  strain  with 
the  lapse  of  time  after  the  lircit  of  elasticity  has  been  consider- 
ably passed. 

Values  of  the  coefficient  of  elasticity,  therefore,  mean  little 
after  that  limit  is  exceeded. 

The  results  of  the  experiments  on  bar  No.  I  are  shown 
graphically  in  Fig.  I.  The  values  of  "/  "  are  laid  off  vertically 
through  O  to  a  scale  of  20,000  pounds  to  the  inch  ;  the  tensile 
strains  are  the  horizontal  co-ordinates  of  the  curve  laid  down 
at  full  size.  The  essentially  straight  portion  of  the  curve 
between  O  and  a  is  within  what  is  ordinarily  known  as  the 
"  elastic  limit." 

The  equation  for  this  portion  of  the  line  is  : 


E  being  assumed  constant  if  Oa  is  considered  straight. 


220  WROUGHT  IRON  IN  TENSION.  [Art.  32, 


The  point  a  is  at  a  vertical  distance  above  O  indicating 
about  24,000  pounds  per  square  inch,  i.e.9  about  the  elastic  limit. 
Above  this  point  the  curvature  of  the  line  is  very  sharp,  indicat- 
ing a  rapid  fall  in  the  value  of  E  and  a  rapid  rise  in  the  values 
of  the  strains  /or  LI.  For  "/  "  =  27,000  (nearly)  the  table  shows 
E  ~  23,000,000  (nearly)  and  LI  —  0.12  inch  ;  while  for  "/"  = 
37,000,  E  =  4,100,000  and  LI  =  1.095  inches  (nearly).  These 
phenomena  are  always  characteristic  of  the  limit  of  elasticity. 

Above  the  point  b  the  curvature  is  slight,  indicating  (what 
the  table  shows)  a  comparatively  slow  change  in  the  values  of  E. 

The  table  shows  that  bar  No.  2  would  exhibit  a  curve  of 
precisely  the  same  character  but  with  a  more  rapid  decrease  to 
E  above  the  elastic  limit.  The  tests  of  this  bar  were  not  car- 
ried to  failure  on  account  of  the  breaking  of  one  of  the  holding 
details. 


UNIVERSITY 


Art.  32.] 


COEFFICIENT  OF  ELASTICITY: 


OF 


Within  the  elastic  limit,  the  mean  values  of  E  may  be  taken 
about  as  follows  : 


For  bar  No.  I  : 


For  bar  No.  2  : 


E  —  28,000,000  pounds. 


E  =  27,500,000  pounds. 


The  next,  Table  IV.,  contains  values  of  the  coefficient  of 
tensile  elasticity  (E)  determined  by  Knut  Styffe  ("  The  Elas- 
ticity, Extensibility  and  Tensile  Strength  of  Iron  and  Steel," 
translated  from  the  Swedish  by  Christer  P.  Sandberg). 


TABLE    IV. 


KIND    OF   IRON. 

PER    CENT 

AREA   OF 

OF 

SET. 

E. 

SECTION. 

CARBON. 

Sq.  inch. 

Inch. 

Pounds. 

Hammered  Bessemer  Iron 

(square) 

.1 

0.1003 

O.OO2 

32,320,020 

.15 

0.1107 

0.001 

34,241,380 

Puddled,  from  Low  Moor 
Dudley 

(round)  .  .     . 

.20 
.09 
.09 

0.1961 
0.184* 

O.2Oo6 

O.OO6 
O.OOS 
0.077 

31,976,920 
28,408,680 
27,448,000 

" 

Motala,  Sweden 

u       

.05 

0.1942 

0.008 

30,261,420 

From  Surahammar 

(square)  „. 

.20 
.14 

o.  1229 
0.2176 

O.OlS 

29.575,220 
31,084,860 

u 

20 

o.  1269 

O.OO2 

30,467,280 

Swedish  Rolled  Iron  from  Aryd 

M 

j  •:  ( 

0.2087 

0.037 

26,761,800 

ii                        11                    il                 fcl                   14 

M 

.18 

0.2279 

O.OO3 

27,791,000 

41      Hallstahammer  (square). 

.07 

0.1891 

0.013 

28,957,640 

0.07 

o.  1905 

O.OOI 

30,810,380 

The  "  Set  "  is  the  permanent  elongation  which  "  the  bar 
had  just  before  the  modulus  (E)  was  taken."  In  the  per  cent, 
of  carbon  no  distinction  is  here  made  between  "  in  the  bar 
tested  "  and  "  in  the  bars  of  the  same  kind,"  the  two  quanti- 
ties given  by  Styffe. 


222  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

As  a  result  of  his  experiments  in  regard  to  the  effect  of  a 
change  of  temperature  on  the  coefficient  of  tensile  elasticity, 
he  states  (page  112  of  the  work  above  cited)  : 

"  That  the  modulus  (coefficient)  of  elasticity  in  both  iron 
and  steel  is  increased  on  reduction  of  temperature  and  dimin- 
ished on  elevation  of  temperature  ;  but  that  these  variations 
never  exceed  .05  per  cent,  for  a  change  of  temperature  of  1.8° 
Fahr.,  and  therefore  such  variations,  at  least  for  ordinary  pur- 
poses, are  of  no  special  importance." 

In  his  "  Physique  Mecanique,"  page  58  of  the  "  Premier 
Me"moire,"  M.  G.  Wertheim  gives  three  coefficients  of  tensile 
elasticity  for  wrought  iron,  each  having  about  the  value  of 
29,680,000  pounds  per  square  inch,  and  one  for  iron  wire  of 
about  26,474,000  pounds  per  square  inch. 

Redtenbacher  (Resultate  fur  den  Maschinenbau,  Zweite 
Auflage,  page  36)  gives  as  the  limits  of  the  values  of  the  co- 
efficient of  elasticity,  expressed  in  pounds  per  sq.  in.,  about 
21,330,000  and  35>55o>ooo- 

Reviewing  the  preceding  values,  therefore,  it  would  appear 
that  the  coefficient  of  tensile  elasticity  for  good  wrought  iron 
may  be  ordinarily  taken  to  lie  between  25,000,000  to  30,000,000 
pounds  per  square  inch,  with  extreme  values  arising  from  vari- 
ation  of  mode  of  manufacture,  chemical  constitution,  size  of 
bar,  etc.,  lying  some  distance  either  side  of  those  limits. 

Since  E  =  —-,  if  /  =  I,  /=  —=r  will  be  the  elongation   or 
/  JbL 

tensile  strain  for  each  unit  of  stress ;  hence,  the  coefficient  of 
elasticity  is  the  reciprocal  of  the  strain  for  a  unit  of  stress*  For 
an  intensity  of  stress  of  20,000  pounds,  for  example,  then  : 

,          20,000  20,000 

/  = to 


25,000,000  30,000,000 


to 


1250  1500 


Art.  32.]  ULTIMATE  RESISTANCE. 

or  a  bar  of  wrought  iron  will  be  stretched  : 


223 


th      to     -I-  th 


1250  1500 

of  its  length. 

The  coefficient  of  elasticity  is  thus  seen  to  be  a  measure  of 
the  stiffness  of  the  material. 


Ultimate  Resistance  and  Elastic  Limit. 

It  has  been  found  by  experiment  that  bars  of  wrought  iron 
which  are  apparently  precisely  alike,  in  every  respect,  except 
in  area  of  normal  section,  do  not  give  the  same  ultimate  tensile 
resistance  per  square  inch.  Other  things  being  the  same,  bars 
of  the  smallest  cras.s~s£ctian-gwe  the -greatest  intensity  of  ultimate 
tensile  resistance. 

Aside  from  the  absence  of  uniform  distribution  of  stress  in 
the  interior  of  the  bar,  as  was  shown  in  the  section  "coefficient 
of  elasticity"  and  the  intensified  effects  of  the  processes  of 
production  on  pieces  with  comparatively  small  cross  sections, 
this  result  is  to  be  expected  from  the  circumstances  which 
attend  fracture.  When  a  piece  of  material  is  subjected  to 
tension  to  the  point  of  rupture,  not  only  a  tensile  strain  of 
essentially  uniform  character,  from  end  to  end,  takes  place,  but 
also  a  very  considerable  local  transverse  strain,  or  contraction, 
at  the  place  of  fracture.  This  latter  manifests  itself  only 
shortly  before  rupture  as  a  short  "  neck  "  in  the  piece.  Now  a 
given  percentage  of  "  local  "  contraction  in  the  case  of  a  large 
section  involves  a  much  larger  absolute  lateral  movement  of 
the  molecules  than  in  the  case  of 'a  small  section.  But  it  is 
evident  that  this  absolute  lateral  movement  will  exert  a  much 
more  potent  influence  toward  severing  the  molecules  suffi- 
ciently for  rupture,  than  the  percentage  of  contraction.  Hence 


224 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


the  degree  of  local  and  lateral  movement,  required  by  rupture, 
will  be  reached  with  a  less  mean  intensity  of  stress  in  the  cases 
of  large  section  than  in  those  of  small  ones.  But  this  is  equiv- 
alent to  a  greater  intensity  of  ultimate  resistance  for  the  small 
sections,  and,  as  has  been  indicated,  this  conclusion  is  verified 
by  experiment. 

The  same  considerations  result  in  the  additional  conclusion 
that,  other  things  being  equal,  the  smaller  sections  will  give  the 
greater  final  contraction.  But  a  greater  intensity  of  ultimate 
resistance  and  greater  final  contraction  involves  a  greater  final 
stretcJi,  with  the  same  length  of  piece. 

These  last  two  conclusions  will  also  be  found  to  be  here- 
after verified  by  experiment. 

Again,  it  is  found  independently  of  the  effects  of  the  pro- 
cesses of  production,  as  might  be  anticipated,  that  the  length 
in  terms  of  the  lateral  dimensions  of  the  test  piece,  within  cer- 
tain limits,  affects  very  perceptibly  the  ultimate  resistance. 

If  a  specimen  of  the  shape  shown  in  Fig.  2  be  broken  by  a 
tensile  stress,  it  will,  of  course,  fail  in  the  reduced  section  MN. 
But  before  fairure  takes  place,  the  reduced  portion  will  be  con- 

D  siderably  elongated  and  the 
M 


normal  section  correspond- 
ingly reduced,  in  conse- 
quence of  the  shearing 
strains  in  the  oblique  planes 
shown  by  the  dotted  lines. 
(See  Arts.  3  and  4.)  When  the  reduced  portion  in  the  vicinity 
of  MN  is  very  short  in  comparison  with  its  lateral  dimensions, 
it  includes  the  whole  of  very  few  of  these  oblique  planes,  if  any 
at  all,  consequently  very  little  movement  of  these  oblique 
layers  over  each  other  can  take  place ;  in  other  words  little 
or  no  reduction  of  section  can  take  place  before  rupture.  In 
this  latter  case,  then,  a  greater  area  of  metal  section  will  offer 
its  resistance  to  the  external  tensile  force,  at  the  instant  of 
failure,  than  in  the  former,  and  a  correspondingly  greater  in- 


Art.  32.]  ULTIMATE  RESISTANCE.  22$ 

tensity  of  ultimate  resistance  will  be  found.  Thus  the  shape 
and  dimensions  of  the  test  piece  will  considerably  influence  the 
ultimate  resistance  and  strains,  as  will  soon  be  shown  by  ex- 
perimental results. 

All  the  preceding  conclusions,  though  given  in  connection 
with  wrought  iron,  are  independent  of  the  nature  of  the  ma- 
terial, and  apply  equally  to  steel  and  cast  iron. 

Since  the  reduction  of  area  of  the  fractured  section  and  the 
elongation  of  the  bar  are  true  measures  of  the  ductility  of  the 
iron,  these  are  or  should  be  always  measured  with  care. 

Table  V.  exhibits  in  a  very  plain  manner  the  decrease  of 
ultimate  tensile  resistance  with  the  increase  of  sectional  area 
of  round  bars  ;  it  is  taken  from  the  "Report«of  the  Committees 
of  the  U.  S.  Board  appointed  to  test  Iron,  Steel  and  other 
Metals,  etc.,"  by  Commander  L.  A.  Beardslee,  U.S.N. 

This  decrease  is  probably  partly  due  to  the  effect  produced 
upon  the  iron  by  the  rolls  as  it  passes  through  them  ;  the 
bars  of  smaller  sections  being  more  "  drawn,"  and  at  a  lower 
temperature  in  consequence  of  the  lesser  mass  cooling  more 
quickly. 

The  notation  of  the  table  is  the  following  : 


=  diameter  of  the  round  bar  in  inches  ; 
"  71."         =  ultimate  tensile  resistance  ; 
"  E.  L."  =  elastic  limit. 

It  will  be  observed  that  the  ultimate  resistance  per  square 
inch  varies  between  widely  separated  limits,  in  some  cases,  for 
the  same  diameter  of  bar.  This  is  due  to  the  fact  that  the 
different  bars,  even  of  the  same  diameter,  were  from  a  number 
of  different  mills,  and  consequently  involved  different  treat- 
ment in  manufacture,  chemical  constitution,  etc.  A  general 
view  of  the  table,  however,  shows  in  a  marked  and  satisfactory 
manner  the  decrease  of  Twith  the  increase  of  the  diameter  or 
area  of  normal  section.  The  last  fourteen  bars  of  the  table 


226 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


are  of  the  same  manufacture,  and  show  a  decrease   in    T  as 
nearly  uniform  as  could  be  expected. 


TABLE    V. 


Ultimate  Resistance  and  Elastic  Limit  in  Pounds  per  square  inch  of 
Original  Normal  Section. 


DlA. 

T. 

E.L. 

DlA. 

T. 

E.  L. 

DlA. 

T. 

E.L. 

I 

59,885 
54,090 
62,7OO 

40,980 

* 

53»°l6 
51,296 
50,594 

35,379 
34,940 

't* 

50,969 

50,307 
48,953 

30,814 
29,767 

44 

59,000 
57,7oo 



1/2 

57,052 
56,505 

38,417 
32,496 

*y 

55,8o3 
53,100 

31,031 

32,074 

ii 

55,4oo 
52,275 
55,450 

39,126 

54^540 

33,77! 
32,869 

; 

52,875 
52,505 
5M59 

35,641 
32,312 
27,816 

" 

52,050 



54,354 

34,6i7 

< 

50,3D3 



K 

57,66o 
5^546 
50,630 
61,727 

35,933 
33,93i 

S4.-544 
53,512 
52,819 
52,736 

33,027 

34,840 
34,901 

2.0 

5T,°39 
49,744 
48,670 
60,213 

33,067 
35,6i5 
23,250 
31,441 

44 

57,363 

37,4i5 

52,700 

35.88o 

oi  108 

44 

39,230 

52,155 

27,708 

4 

4Q,  164 

j  *,  Ay0 

11 

56,790 

36,885 

5i,994 

32,054 

' 

51,684 

33,!°4 

5J,921 

3Ii300 

5M56 

34-591 

' 

52,127 

32,461 

1/8 

52,819 
51,400 
60,458 
57,470 
57,498 

32,267 
34,6oo 

37,344 
31,900 

15/8 

56,344 
57,402 
56,227 
54,334 

35^89 
35,7oi 
33,207 
32,163 

44 

52,011 

50,000 

50,17! 
47,812 

34,702 
28,567 

36,  T  84 

28,983 
35,864 

55,927 

37j25O 

53,339 

33,540 

44 

48  2J.Q 

|« 

54,644 
53,900 

34,695 
26,787 

53,6i4 
52,675 

30,664 
33,745 

2'' 

>  ^y 
46,15! 

36,050 

" 

53,035 

34,410 

29,364 

1? 

5!,559 



*x 

52,267 
59,46i 

32,019 
36,501 

52,401 
51,205 

34,012 

2# 

49,422 
50,481 



44 

57,897 
55,782 
56,334 

32,469 
35,596 
33,921 

4 

50,970 

56,595 
54,1  14 

33^25 
38,310 

4 

51,225 
48,382 

51,666 

30,459 

44 

55,253 
53,893 

34,784 
32,712 

-j 

57,789 
57,874 

34,160 

*« 

49,290 

32,163 

44 
44 

53,247 
53,752 
52,970 

32,520 
32,075 

54,410 
53,846 
55,018 

3!,354 
36,573 
34,283 

4 

48,898 
46,866 
48,475 

28,241 

28,932 

" 

53,022 



53,264 

47,428 

29,941 

44 
<4 

50,040 
58,926 
58,021 
54,949 
54,277 
52,733 

53,557 
52,537 

30,730 
37,548 
32,152 

33,622 
34,606  • 
33,650 
34,469 

4't 

53,154 
51,509 
50,395 
50,547 
49,816 
50,129 

56,577 

35,323 
29,404 

36,254 
35,954 

32!27i 

3-° 

3F 

\K 
4.o 

47,344 
46,446 

47,014 
47,000 
46,667 
46,322 

29,758 

26,333 

26,400 

24,59! 
24,961 
23,636 
23,430 

In  the  words  of  the  Report,  as  given  by  Wm.  Kent,  C.E.,  in 


Art.  32.] 


ULTIMATE  RESISTANCE. 


227 


the  abridgment,  "  The  elastic  limit  as  given  is  not  from  per- 
fectly accurate  data  ;  it  is  simply  the  amount  of  stress  which 
produced  the  first  perceptible  change  of  form,  divided  by  the 
bar's  area." 


TABLE    Va. 
Rectangular  Bars. 


SIZE  OF 

STRESS    IN    LB 

S.  PER   SQ.   IN. 

PER   CE 

NT.   OF 

NO. 

KIND    OF   IRON. 

BAR. 

Elastic 
Limit. 

Ultimate. 

Final  elongat'n 
in  80  inches. 

Final  contrac- 
tion. 

Inches. 

I 

Single  Refined 

3  x   i 

29,000 

52,470 

18.0 

31.0 

2 

Double 

3  x   i 

31,000 

53,550 

16.0 

27.7 

3 

Single 

5  x  ii 

27,330 

50,410 

16.6 

24.1 

4 

Double 

5  x  ii 

27,170 

50,920 

19.0 

25-7 

5 

Single 

3  x   i 

28,330 

48,700 

I3-I 

27.1 

6 

Double 

3  x  i 

29,170 

51,370 

22.2 

35-6 

7 

Single 

5  x  U 

24,830 

49,240 

16.0 

18.1 

8 

Double 

5  x  ii 

27,170 

51,010 

19.7 

29-5 

Table  Va.  shows  the  results  of  some  tests  in  the  U.  S. 
Govt.  machine  during  1881,  at  Watertown,  Mass.  Nos.  i  and 
2  are  means  of  four  tests  ;  the  others  are  means  of  three.  Nos. 
i,  2,  3  and  4  are  for  bars  from  the  Elmira  Iron  and  Steel  Roll- 
ing Mill  Co.;  Nos.  5,  6,  7  and  8  are  from  the  Passaic  Rolling 
Mill  Co.  As  a  rule  the  large  bars  give  the  least  elastic  limit 
and  ultimate  resistance. 

It  is  also  important  to  observe  that  the  double  refined  iron, 
with  two  exceptions,  gives  the  highest  results  of  all  kinds. 

It  appears  from  an  examination  of  the  tables  that  the  elas- 
tic limit  varies,  approximately,  from  a  half  to  two-thirds  the 
ultimate  resistance. 

The  ultimate  resistance,  it  is  to  be  particularly  observed,  is 


228  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

given  in  pounds  per  square  inch  of  original  sectional  area.  On 
account  of  the  reduction  of  the  fractured  section,  the  ultimate 
resistance  should  be  specifically  referred  either  to  its  own  sec- 
tion (to  be  noticed  hereafter)  or  to  the  original  section.  The 
customary  reference  is  to  the  latter,  though  it  is  frequently 
interesting  and  important  to  make  an  accompanying  reference 
to  the  former. 

The  influence  of  the  reduction  of  the  piles  between  the  rolls 
was  next  examined  by  the  same  committee.  It  was  found  that 
the  additional  working  involved  in  the  increased  reduction  of 
the  pile,  as  it  passes  through  the  successive  rolls,  in  the  process 
of  manufacture,  considerably  increases  both  the  ultimate  re- 
sistance and  elastic  limit.  Tables  VI.  and  VII. ,  condensed 
from  those  containing  the  results  of  the  committee's  experi- 
ments, show  this  effect  in  a  very  satisfactory  manner.  The 
notation  is  as  follows  : 

D  —  diameter  of  bar  in  inches  ; 

A  =  area   of  normal  section  of  original  pile  in   square 

inches ; 

Per  cents.  =  area  of  bar  in  per  cent,  of  area  of  pile  ; 
T  —  ultimate  tensile   resistance  in   pounds  per  square 

inch  of  entire  bar  ; 
T'  =  ultimate  tensile  resistance  in  pounds  per  square 

inch  of  core  of  bar ; 
E.  L.  =  elastic  limit  in  pounds  per  square  inch  of  entire 

bar; 
E'.  L'.  =  elastic  limit  in  pounds  per  square  inch  of  core 

of  bar. 

As  is  to  be  anticipated  in  such  cases,  some  irregularities 
are  exhibited  in  the  tables,  but  they  are  very  few,  while  the 
general  result  is  unmistakable.  On  the  whole,  a  considerable 
increase  in  the  values  of  T  is  observed  in  connection  with 


Art.  32.] 


INFLUENCE   OF  ROLLS. 


229 


a  decrease  in  the  values  of  "  Per  cents''     Values  of  the  elastic 
limit  show  greater  irregularities. 


TABLE   VI. 


Comparison  of  the  Reductions  by  the  Rolls,  with   the  Effects  upon 
Tenacity,  and  Elastic  Limit  of  Round  Bars. 


D. 

A. 

PER  CENTS. 

T. 

T'. 

E.  L. 

E'.  L'. 

4 

80 

15-70 



46,322 



23,430 

3i 

80 

12.03 



47,000 



24,961 

3 

80 

8.83 



47,76i 



26,400 

.  2* 

So 

6.13 

47,344 

47,428 

29,758 

29,941 

2 

72 

4-36 

47,872 

48,280 

35,864 

31,892 

If 

36 

6.68 

50,547 

48,792 

35,954 

38,992 

li 

36 

4.90 

50,820 

51,838 

35,o87 

36,467 

I* 

36 

3-4i 

52.729 

49,801 

39,608 

40,534 

I 

25 

3-14 

51,921 

51,128 

39,066 

38,596 

ft 

I2| 

3.60 

50,673 

50,276 

33,933 

35,933 

* 

9 

2.17 

52,275 

52,775 

38,445 

39,126 

4 

3 

i.  60 

57,000 

59,585 

Lost 

Lost 

TABLE   VII. 

Another  Table   showing  Similar  Results,  with   T'  and  E'  L ,  for 

Core,  omitted. 


D. 

A. 

PER  CENTS. 

T. 

E.  L. 

2 

27 

11.63 

51,848 

32,461 

li 

15 

11.78 

53,550 

34,690 

lil 

27 

10.22 

54,034 

33,610 

If 

15 

9.90 

54,277 

33,622 

I| 

27 

8.90 

55,018 

34,283 

Ij 

15 

8.18 

56,478 

33,251 

If 

27 

7.68 

56,344 

35,889 

I* 

15 

6.62 

56,143 

32,267 

The  opinion  of  the  committee  on  the  effect  of  underheating 


230 


WROUGHT  IRON  IN    TENSION. 


[Art.  32. 


or  overheating  is  thus  given  in  the  abridgment  of  their  report 
by  Wm.  Kent,  M.E. :  "  The  indications  are  that  if  a  bar  is 
underheated  it  will  have  an  unduly  high  tenacity  and  elastic 
limit,  and  that  if  overheated  the  reverse  will  be  the  case." 

In  the  words  of  the  report :  "  The  evidence  submitted  is  of 
sufficient  value  to  justify  us  in  asserting  that  variations  in  the 
amount  of  reduction  by  the  rolls  of  different  bars  from  the 
same  material  produce  fully  as  much  difference  in  their  physi- 
cal characteristics  as  is  produced  by  differences  in  their  chemi- 
cal constitution." 

The  committee  also  made  some  valuable  experimental  in- 
vestigations with  the  object  of  ascertaining  the  influence  of  the 


I        J 

t     5r 

a 

c 

a 

b 

\              I 

/               \ 

C         1 

Fig.  3 


Fig.  4 


relative  dimensions  of  the  test  piece,  already  remarked  upon  in 
connection  with  Fig.  2.  Eighteen  specimens  were  prepared, 
of  which  Figs.  3  and  4  represent  types. 

Fig.  3  represents  a  specimen  whose  middle  portion  is  turned 
down  to  a  uniform  diameter.  Seventeen  of  the  specimens 
were  of  this  kind,  with  lengths  of  cylindrical  portions  varying 
from  T/2,  inch  to  10  inches.  Fig.  4  represents  the  eighteenth 
specimen  with  simply  a  groove  in  the  centre,  in  which,  at  ab, 
the  fracture  took  place.  In  this  latter  specimen,  the  reduction 


Art.  32.] 


INFLUENCE   OF  LENGTH. 


231 


of  area  at  the  section  of  failure  must  necessarily  be  much  less 
than  in  those  like  Fig.  3  ;  hence,  the  ultimate  resistance  will  be 
correspondingly  greater. 

Table  VIII.  is  taken  from  the  report  already  cited,  and 
contains  the  results  of  the  experiments  on  the  eighteen  speci- 
mens prepared  in  the  manner  indicated  above. 

L  =  original  length  in  inches  ; 
/  —  per  cent,  of  elongation  ; 
a  —  per  cent,  of  contraction  of  fractured  area  ; 
t  =  stress  in  pounds  per  square  inch  at  first  stretch  ; 
T  =  ultimate  tensile  resistance   in  pounds  per  square 
inch  of  original  section. 


TABLE   VIII. 


NO. 

L. 

* 

a. 

t. 

T. 

REMARKS. 

I 

10 

23.1 

38.2 

29,678 

54,888 

Slight  seam. 

2 

9i 

24-3 

36.5 

28,011 

55,288 

3 

9 

21  5 

31-1 

29,345 

55,355 

4 

8* 

22.  0 

31-2 

29-345 

55-622 

5 

25-0 

39-9 

30,840 

54.890 

Slight  seam. 

6 

7 

25-8 

38.6 

30,412 

55,488 

7 

22.  I 

40.0 

28,562 

51,800 

Bad  seam. 

8 

6 

22-3 

34-7 

30,600 

55,  4l8 

9 

10 

Si 

5 

25-4 
21.2 

39-3 
32.2 

29,475 
29,278 

55,333 
55,887 

Slight  seam. 

ii 

4 

25-7 

37-4 

29.705 

55,532 

12 

3i 

26.7 

36.6 

31,817 

55,482 

13 
14 

3 

2 

27.0 
27.0 

38-3 
36.2 

31,123 
33,428 

56,190 
56,428 

Seamy. 

15 

li 

26.O 

34-o 

42,249 

57.096 

ft 

16 
17 

I 

37-o 
30.0 

34.3 
37-9 

34,288 
57,565 

58.933 
59,388 

Seamy. 

18 

Groove 

~""~~l 

20.  6 

45,442 

7J,300 

The  diameters  at  the  section  of  failure  were  nearly  uniform 
and  originally  about  0.97  inch. 


232  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

The  values  of  /,  a  and  T  are  as  nearly  uniform  as  could  be 
expected  until  the  length  decreases  to  about  4  diameters 
(2  inches). 

For  the  grooved  specimen  t  and  T  are  very  large,  and  a 
very  small. 

Other  experiments  on  a  still  softer  iron  were  made  with 
the  same  general  results. 

"  In  conclusion,"  states  the  committee,  "  our  results  lead 
us  to  the  decision  that,  in  testing  iron,  no  test  piece  should  be 
less  than  one  half  inch  in  diameter,  as  inaccuracy  is  more  prob- 
able with  a  small  than  with  a  large  piece,  and  the  errors  are 
more  increased  by  reduction  to  the  square  inch  ;  that  the 
length  should  not  be  less  than  four  times  the  diameter  in  any 
case  ;  and  that,  with  soft,  ductile  metal,  five  or  six  diameters 
would  be  preferable." 

In  Vol.  II.  of  the  a  Transactions  of  the  American  Society  of 
Civil  Engineers,"  Mr.  C.  B.  Richards  has  given  a  paper  in 
which  are  recorded  the  results  of  some  experiments  exhibiting 
the  influence  of  the  relative  dimensions  of  the  specimens.  The 
average  of  eight  tests  of  Burden's  "  best "  iron,  with  "long" 
specimens  (similar  to  Fig.  3)  varying  from  5  to  5^J  inches  in 
length  and  0.62  to  i.oo  inch  in  diameter,  gave  : 

T  =  49,588  pounds  ;          a  —  4.6.7  per  cent.  ; 
/  =  30.4  per  cent. 

With  "  short "  specimens  (like  Fig.  4)  of  the  same  iron,  the 
average  of  six  tests  gave  : 

T  —  62,089  pounds  ;  a  —  29.5  per  cent. 

The  large  value  of  T  and  small  value  of  a,  for  the  "  short  " 
specimens,  are  thus  seen  to  be  very  marked  in  contrast  with 
the  same  quantities  for  the  "long"  specimens. 


Art.  32.]  INFLUENCE   OF  SKIN.  233 

Other  experiments  of  Mr.  Richards,  showing  the  same  re- 
sults, will  be  given  in  connection  with  the  resistance  of  boiler 
plates. 

It  has  long  been  the  impression  that  there  exists  a  consid- 
erable difference  between  the  ultimate  tensile  resistance  of  the 
"  skin  "  of  a  bar  of  iron  and  that  of  the  portion  of  the  bar  un- 
derneath the  skin.  The  U.  S.  committees,  therefore,  broke  a 
number  of  bars  first  with  the  skin  on,  or  "  in  the  rough,"  and 
then  with  the  skin  turned  off.  In  a  large  majority  of  the  cases, 
the  rough  bars  gave  the  highest  ultimate  resistance  per  square 
inch,  by  a  small  amount,  while  in  a  few  cases  the  results  were 
of  the  opposite  character.  On  the  whole,  however,  "  the  ac- 
cumulated evidence  indicates  that  the  strength  of  the  skin  of 
the  bar  is  greater  in  proportion  to  its  area  than  that  of  the  rest 
of  the  bar." 

All  the  tests,  of  which  the  results  have  hitherto  been  given, 
were  made  on  round  bars,  or  on  specimens  turned  from  them. 
Results  of  tests  on  other  iron  will  now  be  detailed,  and  it  will 
be  convenient  to  use  the  following  and  customary  symbols  for 
the  various  kinds  of  "  shape"  irons  : 

L  ,  for  angle  irons  : 

JL  ,  for  tee  irons  ; 

C  ,  for  channel  bars  ; 

I  ,  for  eye  beams  ; 

P  ,  for  rectangular  bars  or  "  flats  "  ; 

O  ,  for  rounds  ; 

-(•, ,  for  star  sections  ; 

in  short,  any  shape  iron,  or  steel,  is  represented  by  a  skeleton 
of  its  section. 

Table  VI 11^.  contains  the  results  of  tests  by  Henry  Wood, 
Inspector  for  the  Q.  M.  O.  &  O.  Railway  of  Canada,  on  some 
bridge  members  at  Phcenixville,  Penn.,  in  Oct.,  1878. 


234 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


TABLE   Villa. 


SECTION. 

LENGTH. 

E.L, 

r. 

CONT. 

STRAIN. 

REMARKS. 

Inches. 

Inches. 

3i  x   ITS 

338 

31,000 

56,414 

0.28 

0.14 

3*  x     g 

264 

32,000 

54,444 

0.40 

0.15 

Flats. 

3     x   ij 

336 

29,000 

6i,33i 

O.Og 

0-.  II 

Not  broken. 

3*  x     fj 

80 

30,000 

56,474 

0.27 

0.15 

ifo 

320 

31,900 

57,530 

0.30 

0.13 

1^0 

80 

31,800 

56,454 

0-43 

0.19 

Rounds. 

Ifo 

254 

27,000 

56,351 

O.5O 

0.17 

E.  L.  =  elastic  limit  in  pounds  per  square  inch. 

T.  =  ultimate  resistance  in  pounds  per  square  inch  of 

original  section. 

Cent.  —  reduction  of  original  area. 
Strain  =  stretch  of  "  Length." 


The  iron  in  these  bars  was  that  of  the  Phoenix  Iron  Co. 

It  should  be  stated  that  the  tests  were  made  in  a  lever 
machine  in  which  the  friction  may  have  amounted  to  4  or  5 
per  cent,  of  T. 

The  following,  Table  IX.,  contains  the  results  of  some  ex- 
periments made  in  1875  by  Mr.  G.-Bouscaren  for  the  Trustees 
of  the  Cincinnati  Southern  Railway,  and  is  taken  from  the 
"  Report  on  the  Progress  of  Work,  etc.,"  by  Thomas  D.  Lovett, 
Con.  and  Prin.  Engr.,  published  in  1875. 

The  tests  were  made  on  eye  bars,  and  only  such  have  been 
selected  for  the  table  as  broke  in  the  body  of  the  bar  and  at  a 
section  out  of  reach  of  the  influence  of  the  fire  used  in  the 
process  of  forming  the  head. 

Nos.  2  to  1 8  are  corrected  for  the  error  of  a  spring  gauge, 
as  is  indicated  in  the  report  cited. 


Art.  32.] 


EYE  BARS. 


235 


TABLE   IX. 


NO. 

BRAND. 

LENGTH. 

SECTION. 

r. 

STRETCH. 

Ft.  In. 

Inches. 

Per  cent. 

2 

D.  S. 

10  9ft 

0      y     J-L 

3    x   to 

50,667 

20 

6 

P. 

10  gft 

3   x  £ 

47,734 

7 

8 

D.  S. 

5  o 

3   x.* 

50,220 

10 

9 

14 

5  o 

3  x  | 

50,220 

13 

10 

i  i 

5  o 

3  x  f 

50,220 

14 

ii 

*4 

5  o 

3  x  £ 

48,980 

II 

12 

4  < 

5  o 

3  x  3 

51,556 

14 

13 

*  * 

5  o 

3  x  3 

48,980 

10 

14 

P. 

5  o 

3  x  i 

53,870 

19 

15 

'' 

5  o 

3  x  i 

53,S7o 

18 

16 

11 

5  o 

3  x  i 

49,5oo 

16 

17 

4< 

5  o 

3  x  i 

50,666 

15 

18 

" 

5  o 

3   x  i 

47,38o 

ii 

26 

" 

5  o 

3  x  4 

53-700 

16 

29 

D.  S. 

5  o 

3  x  f 

52,600 

13 

30 

M 

5  o 

3  x  i 

50,600 

13 

3i 

" 

5  o 

3  x  f 

52,600 

14 

32 

4  * 

5  o 

3  x  | 

48,100 

10 

33 

" 

5  o 

3  x  i 

50,600 

15 

35 

P. 

5  o 

3  x  | 

52,000 

15 

36 

" 

5  o 

3  x  f 

52,900 

14 

37 

" 

5  o 

3  x  f 

50,900 

7 

38 

1  ' 

5  o 

3  x  £ 

55,ioo 

13 

40 

D.  S. 

5  o 

3  x  f 

48,100 

14 

41 

11 

5  o 

3  x  f 

42,400 

14 

42 

" 

5  o 

3  x  f 

43,800 

17 

43 

*  * 

5  o 

3  x  f 

49,800 

15 

45 

*  * 

5  o 

3  x  | 

49,200 

16 

48 

" 

5  o 

3  x  4 

43,900 

15 

53 

5  o 

3  x  f 

47,500 

16 

The  following  is  the  notation  : 

{D.  S.  represents   Diamond  State  Works, 
Wilmington,  Del. 
P.   represents  Pencoyd  Works,  Philadel- 
phia, Penn. 
"  Length  "  —  distance  from  centre  to  centre  of  eyes. 


236  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

"  Section  "  represents  dimensions  in  inches  of  rectangu- 
lar section  of  bars. 

T.  =  ultimate  tensile  resistance  in  pounds  per  square 
inch  of  original  section. 

"  Stretch  "  represents  per  cent,  of  original  length. 

Table  X.  has  been  written  by  the  aid  of  the  same  report  as 
that  from  which  the  preceding  was  taken. 

It  contains  the  records  of  tests  made  on  specimens  sup- 
plied, under  specifications,  by  various  bridge  building  com- 
panies from  the  mills  indicated  in  the  column  "Brand"  With 
the  exceptions  of  Nos.  15,  1 6  and  17,  the  test  specimens  were 
turned  I  j{  inches  round  ;  those  three  were  of  rectangular  sec- 
tion and  of  the  dimensions  given  in  the  column  "  Diam" 

It  is  stated  in  the  report  that  the  ultimate  resistances  are  a 
little  too  large,  as  they  involve  the  friction  of  the  plunger  in 
the  hydraulic  cylinder  of  the  testing  machine. 

The  notation  is  the  following  : 

f  "  P."  Phoenix  Iron  Co. ; 
"  U.  /.,"  Union  Iron  Mills  ; 
"  C.  R."  Cleveland  Rolling  Mills  ; 
<  "  O.F.,"  Ohio  Falls  Iron  Works  ; 
"  D.  S."  Diamond  State  Works  ; 
"  K.  I."  Kellogg  Iron  Works. 
"Dia."  =  diameter,  or  dimensions,  in  inches,  of  original 

section. 
"  T."  =  ultimate  tensile  resistance  in  pounds  per  square 

inch  of  original  section. 

11  Strain  "  =  elongation  for  eight  inches  in  length. 
"  Cent"  =  contraction  of  original  section. 

The  two  latter  were  measured  after  failure  took  place  ;  they 
have  been  computed  from  the  data  given  in  the  report. 


Art.  32.] 


SPECIMEN   TESTS. 


237 


TABLE   X. 

Tests  of  Specimen  Bars  of  Iron,  Submitted  with  Bids  for  Ohio  River 

Bridge. 


NO. 

BRAND. 

DIA. 

T. 

STRAIN. 

CONT. 

REMARKS. 

I 

P. 

•25 

54,400 

0.31 

o-45 

i   I  bar  bent  180°  without 

2 

P. 

•25 

54-400 

O.29 

0.41 

fracture. 

3 

U.    I. 

•25 

53,800 

0.31 

o  40 

2  bars  bent  180°  with- 

4 

U.  I. 

•25 

53,800 

0.28 

o-37 

out  fracture. 

5 

K.  I. 

.24 

54,600 

O.3O 

0-39 

)  i  bar  bent  180°  without 

6 

K.  I. 

•25 

53,400 

O.29 

0.41 

\      fracture. 

7 



•25 

55,ioo 

0.21 

O.2O 

8 



•25 

55,500 

0.31 

0.38 

9 

C.  R. 

•25 

53-800 

0.17 

0.15 

10 

C.  R. 

•25 

54,200 

O.ig 

0.21 

{i  bar  bent  180°  without 

ii 

12 

C.  R. 
C.  R. 

.24 
.24 

60,800 
60,800 

0.28 
O.266 

0-31 
0.27 

fracture. 
I  bar  bent  135°.     Frac- 
ture    showed    partly 

crystalline. 

13 
14 

U.  I. 

U.  I. 

1.25 
1.25 

52,000 
50,400 

0.27 
0.28 

0.36 
0-33 

j-  i  bar  bent  135°. 

(2.59) 

15 

U.  I. 

x 

51,700 



0.08 

(0.49! 

(2-59 

16 

U.  I. 

]        X          ' 

49,400 



O.Og 

(0.48 

2.64 

17 

U.  I. 

]     x      - 

52,400 



O.II 

(o.37 

18 

19 
20 

O.  F. 
0.  F. 
D.  S. 

1.25 
1.25 
1.25 

51,600 
52,200 
53>ooo 

0.13 
0.13 
0.30 

O.Og 
O.Og 
0.43 

}2  bars  bent  135°. 
2  bars  bent  180°  with- 

21 

D.  S. 

1-25 

52,100 

0.28 

O.42 

J      out  fracture. 

In  regard  to  the  column  of  remarks  the  report  says:  "  One 
or  two  of  the  specimen  bars  furnished  by  each  bidder  were 
tested  by  being  bent  cold  under  the  hammer  until  fracture,  and 
the  result  is  noted  in  tfie  column  '  Remarks.' ' 

These  tests  were  made  in  1875. 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


The  next  table  is  again  taken  from  the  same  report,  and 
contains  the  results  of  tests  made  on  specimens  taken  from 
plates,  angles,  rods,  Keystone  column  iron  and  rivet  iron.  "7"," 
as  usual,  represents  the  ultimate  tensile  resistance  in  pounds 
per  square  inch  of  original  section. 

TABLE   XL 


NO. 

SPECIMEN. 

T. 

REMARKS. 

Inches. 

I 

0.86  x  0.40 

50,  ioo 

Specimen  from  14!  x  f  plate. 

2 

0.88  x  0.40 

48,900 

I4l  x  | 

3 

0.86  x  0.36 

45,400 

10    x  i 

4 

0.84  x  0.38 

4I,4OO 

10       X    f 

5 

0.86  x  0.26 

49,  200 

IO       X    4 

6 

0.90  x  0.26 

44,400 

10     x  i 

7 

0.62  x  0.38 

56,000 

10       X    g 

8 

0.64  x  o  36 

49,900 

10     x  § 

9 

0.62   x  0.40 

54»  600 

14*  x  i 

10 

0.62   x  0.42 

52,600 

14!  x  f 

ii 

0.84  x  0.40 

52,600 

2j  x   2j  angle  iron. 

12 

13 

0.54  x  0.42 

Ti5.    v     T 
1|  ti      X      1 

53,800 
46,  700 

24    X    24        "          " 
lib    x   i  bar. 

14 

if  Diam. 

55,2oo 

if  rod. 

15 

ii      " 

58,100 

1  8  rod. 

16 

0.84  x  0.26 

47,700 

6'    Keystone  Column. 

17 

0.84  x  0.30 

49,000 

8" 

18 

1  Diam, 

48,200 

|"  rivet  for  column. 

19 

i 

46,900 

r  "   " 

Nos.  i,  3,  5,  7  and  9  were  taken  from  the  edges  of  the 
plates,  while  Nos.  2,  4,  6,  8  and  10  were  taken  from  the  middle 
of  the  same.  It  is  important  to  observe  that  the  specimens 
from  the  edge,  in  every  instance,  gave  considerably  higher 
values  of  T  than  those  from  the  middle. 

These  tests  were  also  made  in  1875,  and  as  they  were  com- 
pleted in  a  Riehle  machine,  they  may  be  confidently  considered 
exact.  All  the  iron  was  rolled  at  the  Union  Iron  Mills  of  Pitts- 
burgh, Penn. 


Art.  32.] 


PLATE   SPECIMENS. 


239 


The  same  report  will  again  be  drawn  on  for  Table  XII.  of 
"  Tests  of  Tensile  Strength  of  Iron  furnished  (in  1875)  by  the 
Baltimore  Bridge  Co.,  for  Kentucky  River  Bridge." 


TABLE   XII. 


p 

•f. 

BRAND. 

D. 

T. 

1 

STRAIN. 

H 

8 

REMARKS. 

I 

w.  w. 

0.88     x  0.26 

48,500 

O.II 

0.25 

Spec 

men  from  edge      of  12  x  £  plate. 

2 

1  ' 

0.98     x  0.26 

49,100 

O.II 

0.17 

middle 

12    X    4 

3 

" 

0-75     x  0.50 

47,700 

0.17 

O.22 

edge 

12    X    $ 

4 

<  < 

0.75     x  0.50 

46,900 

O.II 

0.12 

edge 

12    X   4 

5 

14 

0.75     x  0.50 

45,300 

0.10 

0.22 

middle 

12    X    | 

6 

U.  1. 

0-375  x  0.75 

50,100 

0.18 

0.12 

edge 

16  x  | 

7 

" 

0-375   x  0.75 

54,400 

O.  12 

0.16 

middle 

16  x  | 

8 

w.  w. 

0.80     x  0.50 

46,000 

0.21 

0.22 

edge 

16  x  I 

9 

" 

0.72     x  0.52 

42,900 

0.07 

0.13 

middle 

16  x  }  . 

10 

(  1 

0.75     x  0.50 

46,400 

0.17 

0.22 

edge 

16  x  4 

ii 

14 

0.75     x  0.48 

48,800 

0.21 

0.21 

middle 

16  x  4 

12 

U.  I. 

0.75     x  0.50 

49.500 





edge 

16  x  | 

13 

" 

0.75     x  0.50 

48,000 



middle 

16  x  4 

14 

" 

0.75     x  0.50 

45,300 

0.10 

0.18 

edge 

18  x  i 

T  r 

<  i 

o.  75     x  o.  50 

A  1    C\C\C\ 



middle 

18  x  4 

A  D 

16 
17 

M 

i.oo     x  0.31 
i.oo     x  0.31 

4/  1  y*-*-' 

49,500 
50,600 

0.16 
0.14 

0.29 
0.26 

edge 
middle 

19  x  -ft 
19  x  A 

18 
19 

" 

i.oo     x  0.25 
i.oo     x  0.25 

56,000 
56,500 

0.13 
0.09 

0.17 
0.08 

edge 
middle 

24  x  4 
24  x  i 

The  notation  is  as  follows  : 


"Brand": 


"  W.  W.;'  Wilson  Walker  &  Co.,  of  Pitts- 
burgh, Penn. 
"  U.  /.,"  Union  Iron  Mills  of  Pittsburgh, 

Penn. 

"  D"  =  dimensions  of  sections  of  specimens  in  inches. 
"  T."  —  ultimate  tensile  resistance  in  pounds  per  square 
inch  of  original  section. 


240 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


"  Strain  "  =  elongation,  at   failure,  for  three   inches  of 

length. 
"  Cont."  —  contraction,  at  failure,  of  original  section. 

Nos.  1 8  and  19  were  taken  from  the  middle  of  the  plate,  but 
all  the  others  were  taken  from  "  crop  "  ends. 

The  report  states  that  the  percentages  are  only  approxi- 
mate, as  they  are  based  on  measurements  made  by  an  ordinary 
rule.  It  also  states  that  specimens  showed  fracture,  on  bend- 
ing cold,  between  90°  and  180°. 

It  is  to  be  observed  that  the  greater  resistance  shown  by 
the  specimens  taken  from  the  edges  of  the  plates,  in  the  pre- 
ceding table,  does  not,  as  a  rule,  appear  in  this  one. 

It  is  inferred,  though  not  definitely  stated  in  the  report, 
that  all  specimens  of  plates  were  tested  in  the  direction  of  the 
fibre,  that  being  the  direction  in  which  the  actual  plates  would 
be  directly  stressed  in  the  structures  contemplated. 

TABLE    XIII. 


SIZE    OF   BAR. 

DIAMETER. 

r. 

E.  L. 

STRAIN. 

CONT. 

Inches. 

Inches. 

4     x   if 

1.  00 

56,200 

31,600 

0.26 

o-33 

4     x   if 

1.  00 

54,600 

26,700 

0.25 

0.31 

4i  x  Jo 

1.  00 

61,100 

31,600 

0.24 

0.28 

4i  x:i| 

1.  00 

59,500 

30,000 

0.17 

0.28 

4i  x  if 

1.  00 

56,600 

27,000 

0.29 

0.31 

4£  x  if 

1.  00 

56,200 

30,000 

O.2O 

0.36 

6     x  i* 

1.  00 

56,200 

28,300 

0.32 

0.41 

6     x  ii 

i.Op 

28,326 

Table  XIII.  contains  the  results  of  tests  made  by  the  writer 
at  Phcenixville,  Penn.,  in  1881.  The  test  specimens  were  all 
one  inch  in  di-ameter  and  turned  to  a  uniform  section  for  a 
length  of  ten  inches  ;  they  were  taken  from  Phcenix  "  Best 


Art.  32.]  BOILER  PLATE.  24! 

Best  "  bars  of  the  sizes  given  in  the  extreme  left  vertical  col- 
umn. The  column  "  E.  L.,"  elastic  limit  in  pounds  per  square 
inch  of  original  area,  contains  values  based  on  measurements 
made  as  accurately  as  possible  with  a  pair  of  fine  dividers  and 
a  scale  graduated  to  hundredths  of  an  inch.  The  results, 
therefore,  can  only  be  considered  as  loosely  approximate. 
The  remaining  notation  is  same  as  used  in  the  preceding 
tables. 

The  failure  of  the  clamps  to  hold  the  last  specimen  pre- 
vented a  complete  record. 

The  "  Strains  "  are  for  six  inches  of  length. 

It  should  be  stated  that  the  testing  machine  was  a  lever 
one,  and  the  friction  may  have  amounted  to  five  per  cent. 

In  all  the  preceding  tables  the  length  for  which  the  "Strain  " 
is  given  should  be  carefully  borne  in  mind. '  A  considerable 
"  local  "  strain  takes  place  at  the  section  of  fracture,  which 
causes  the  per  cent,  of  elongation,  or  strain,  to  be  much  greater 
for  a  very  short  length  than  for  a  longer  one. 


Wrought-Iron  Boiler  Plate. 

The  "  Report  of  the  Committee  of  the  Franklin  Institute," 
made  in  1837,  has  already  been  cited  on  page  215.  That  report 
contains  the  results  of  a  great  number  of  experiments  made  on 
boiler  plates  of  both  wrought  iron  and  copper.  A  very  few 
of  the  values  of  the  ultimate  tensile  resistance,  7",  in  pounds 
per  square  inch,  are  given  in  Table  II.  of  this  Article,  and 
reference  may  be  made  to  it. 

Table  XIV.  contains  the  limiting  values  of  T  from  the 
tables  of  the  report  whose  numbers  are  given  in  the  extreme 
left  hand  column.  The  tables  selected  are  such  as  to  give 
fairly  representative  results,  and  are  chosen  for  no  other 

reason. 

16 


242 


WROUGHT  IRON  IN    TENSION. 


[Art.  32. 


TABLE   XIV. 


TABLE. 

KIND. 

LIMITING   VALUES   OF    T. 

TEMPERATURE. 

FIBRE. 

XXXI. 

H. 

53,543.00  to  66,500.00 

About    65°  Fahr. 

Across. 

XXXII. 

H. 

34,990.00  to  48,041.00 

"        75° 

" 

XXXVI. 

P. 

48,308.00  to  73,385.00 

54°  to  580° 

Along. 

XLVII. 



54,361.00  to   78,000.00 

48°  to    88° 

" 

LII. 

English. 

44,149.00  to  65,897.00 

80°  to  580° 

" 

XL. 

H. 

41,734.00  to  65,141.00 

66°  to    90° 

ti 

L. 

Rolled. 

56,869.00  to  65,700.00 

About    70° 

Across. 

LI. 

54,442.00  to  62,709.00 

70° 

"//"."  signifies  "hammered  into  slabs  and  rolled." 
"  P"  signifies  "  puddled  and  rolled." 

In  Table  XXXVI.  the  highest  values  of  T  occurred  at  the 
highest  temperatures. 

In  Table  LII. .the  lowest  values  of  T  occurred  at  lowest 
temperatures. 

The  results  in  Table  XIV.  were  recorded  from  the  tests  of 
"  long  "  specimens  but  of  very  small  area  of  normal  section, 
t.e.9  from  about  o.io  square  inch  to  0.20  square  inch. 

This  committee  made  numerous  experiments  to  determine 
the  resistance  of  boiler  plate  in  different  directions  in  reference 
to  the  fibre  of  the  iron.  The  results  were  by  no  means  of  a 
uniform  character.  In  one  set  of  forty  strips  cut  in  each  direc- 
tion (along  the  fibre  and  across  it),  the  length  strips  showed  an 
excess  of  resistance  varying  from  one  per  cent,  to  twenty.  This 
comparison  was  made  principally  on  the  minimum  resistance 
of  each  bar,  but  the  committee  state  that  the  result  would  not 
have  been  much  different  if  the  mean  had  been  taken. 

On  reviewing  all  their  experiments,  the  committee  con- 
cluded that  lengtJnvise  of  the  fibre,  the  boiler  iron  which  they 
tested  was  about  six  per  cent,  stronger  than  across  the  fibre. 


Art.  32.] 


BOILER  PLATE. 


243 


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244  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

They  also  determined  that  the  weakest  direction  of  all  was 
diagonally  across  the  fibres,  but  their  experiments  did  not  en- 
able them  to  determine  quantitative  results. 

Table  XIV<z.  is  taken  from  the  "Transactions  of  the  Ameri- 
can Society  of  Civil  Engineers,"  Vol.  II.  It  contains  the  re- 
sults of  some  experiments  on  several  different  kinds  of  plate 
iron  by  C.  B.  Richards,  M.E.,  and  among  other  things  it  reveals 
the  difference  between  "long"  and  "  short"  specimens. 

Column  "No"  shows  the  number  of  tests,  of  which  T  is 
the  average  ultimate  tensile  resistance  in  pounds 
per  square  inch,  T'  the  highest  and  Tt  the  lowest, 
all  being  referred  to  the  original  section. 

Column  "  Cont"  shows  per  cent,  (of  original  section)  of 
contraction  at  section  of  failure. 

Column  "  Strain  "  shows  per  cent,  (of  original  length)  of 
elongation. 

Column  "Spec."  shows  kind  of  specimen,  i.e.,  "long"  or 
"  short,"  also  direction  of  stress  in  reference  to  fibre  ; 
"ZZ"  signifies  "long  and  along  fibre ;"  "LC" 
"long  and  across  fibre;"  while  "S£"  and  "SC" 
signify  "  short  and  along  "  or  "  across  fibre,"  respect- 
ively. 

In  column  "  Brand"  "  B.  5."  signifies  "  Bay  State;" 
"B.  S.  H"  "Bay  State  Homogeneous  Metal;" 
"  Tr  "Thorneycroft,"  English;  "Pcim"  "Pennsyl- 
vania ;  "  "  S.  F."  "  Sligo  Fire  Box.' 

Different  brands  of  the  same  make,  though  given  by  Mr. 
Richards,  have  been  neglected. 

The  lengths  for  which  the  "  Strains  "  existed  are  not  given, 
although  they  should  be.  The  long  specimens  were  three  or 
four  inches  between  the  shoulders. 

In  his  "  Treatise  on  the  Resistance  of  Materials,"  Prof.  De 


Art.  32.]  INFLUENCE   OF  ANNEALING.  245 

— 0 ; ; — 

V^lsen  Wood  gives  the  following  results  of  some  boilerplate 
tests  at  the  shops  of  the'Camden  and  Amboy  R.  R.  by  Mr.  F. 
B.  Stevens. 

"  Av.  breaking  weight  in  pounds  per  square  inch. . . .  54,123.00 

Highest "  "       ."         "         "         "    57,012.00 

Lowest"  "       "       "         "         "         "    51,813.00" 

The  experiments  of  Sir  Wm.  Fairbairn  on '  English  boiler 
plate  ("  Useful  Information  for  Engineers,  First  Series,"  p.  259) 
along  and  across  the  fibres,  gave  irregular  results,  but  other 
English  experiments  of  Easton  and  Anderson  would  seem  to 
make  the  resistance  across  the  fibres  from  5  to  15  per  cent,  less 
than  that  along  the  fibres. 


Effect  of  Annealing. 

The  Franklin  Institute  Committee  determined  the  effect  of 
annealing,  at  different  temperatures,  on  about  56  specimens  of 
boiler  plate  and  wire  iron.  Table  XV.  is  condensed  from  that 
giving  their  results  on  boiler  plate. 

The  mean  value  of  T  for  five  specimens  of  iron  wire  0.19 
inch  in  diameter,  before  annealing,  was  : 

T  =  73,83o. 

After  annealing  by  heating  to  redness  and  cooling  in  dry 
ashes,  the  mean  of  five  specimens  was  : 

T'  =  58,101. 

After  annealing  at  red  heat  and  quenching  in  water,  the 
mean  of  another  five  specimens  was  : 

T'  =  53,578. 


246 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


TABLE   XV. 


NO. 

T. 

ANNEALING   TEMP.    FAHR. 

T. 

DECREASE 
BY  ANNEALING. 

I 

57,133 

1,037° 

56,678 

•025 

3 

53,774 

1,  111° 

52,186 

.029 

6 

53,i85 

1,159° 

46,212 

.131 

9 

52,040 

1,237° 

44,165 

.151 

12 

48,407 

Bright  welding  heat. 

39,333 

.187 

15 

48,407 

«              «          «< 

38,676 

.2OI 

18 

76,986 

«             <i          n 

50,074 

•349 

T  is  the  ultimate  tensile  resistance  in  pounds  per  square 

inch  at  ordinary  temperatures,  before  annealing. 
T'  is  the  same  after  annealing  and  cooling. 

The  means  of  sets  of  five,  three  and  four  specimens  of  wire 
0.156  inch  in  diameter,  exactly  similarly  treated,  were,  respect- 
ively : 

T—  89,162.  T  =  48,144. 

77  -  50,889. 

The  process  of  annealing  is  thus  seen  to  decrease  the  ulti- 
mate tensile  resistance  a  very  considerable  amount.  In  many 
cases,  however,  this  may  make  the  iron  very  much  more  valu- 
able, since  annealing  renders  it  much  more  ductile.  If  a  struct- 
ure or  machine  is  subject  to  shocks  or  sudden  applications  of 


Art.  32.]  HARDENING  AND   TEMPERATURE.  247 

loading,  a  very  stiff,  hard  iron,  originally  utterly  unfit  for  the 
purpose,  after  being  annealed  might  be  used  in  its  construc- 
tion with  safety. 


Effect  of  Hardening  on  the  Tensile  Resistance  of  Iron  and 

Steel. 

It  has  been  seen  that  annealing  reduces  the  ultimate  resist- 
ance of  wrought  iron.  Experiments  have  shown  that  harden- 
ing, on  the  other  hand,  increases  the  resistance  of  both  iron 
and  steel,  provided  the  hardening  is  done  in  a  proper  manner. 
If  the  hardening  is  accomplished  by  heating  and  sudden  cool- 
ing in  water,  without  subsequent  tempering,  the  resistance  of 
hard  steel  is  very  much  diminished.  This  is  probably  due  to 
the  internal  stresses  induced  by  the  sudden  cooling. 

Knut  Styffe  ("  Iron  and  Steel ")  concluded  from  his  experi- 
ments that  "  by  heating  and  sudden  cooling  (hardening),  the 
limit  of  elasticity  is  raised  while  the  extensibility  is  diminished, 
not  only  in  steel  but  also  in  iron."  TJiis,  results  of  the  experi- 
ments by  David  Kirkaldy  will  be  given  hereafter. 


Variation  of  Tensile  Resistance  with  Increase  of  Temperature. 

Table  XVI.  has  again  been  condensed  from  a  similar  one 
given  in  the  Report  of  the  Franklin  Institute  Committee. 

The  third  column  gives  the  temperatures  at  which  the  ulti- 
mate tensile  resistances  in  the  fourth  column  were  observed. 

The  committee  observed  that  the  resistance  of  many  irons 
increased  with  the  temperature,  to  nearly  the  boiling  point  of 
mercury  in  some  cases,  while  others  remained  unchanged  until 
a  temperature  .of  572°  was  reached.  Above  this  point,  how- 
ever, as  a  rule,  they  found  the  decrease  of  resistance,  below  the 
greatest,  to  vary  about  as  the  2.6  power  of  (Temp.  -  80°). 


248 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


TABLE    XVI. 


ULTIMATE   TENSILE   RESISTANCE 
AT   ORDINARY  TEMPERATURE. 

TEMP.    FAHR. 

ULTIMATE   TENSILE   RESISTANCE 
AT   OBSERVED    TEMPERATURE. 

I 

56,736 

212 

67,939 

5 

62,646 

394 

67,765 

9 

49,782 

440 

59,°85 

13 

52,542 

552 

55,939 

17 

53,385 

562 

59,623 

21 

66,724 

572 

66,620 

25 

76,071 

574 

65,387 

29 

59,234 

576 

66,065 

33 

45,757 

578 

53,465 

37 

•59,530 

630 

_6o,oio 

41 

_  5  2,  542 

732 

"53,375 

45 

59,2I9 

819 

55,892-**-  

49 

59,219 

1,022 

37,4io 

53 

54,768 

1,142 

18,672 

56 

53,426 

I,l87 

21,910 

59 

54,758 

1,317 

18,913 

In  the  London  "  Engineering  "  of  3Oth  July,  1880,  is  given 
a  synopsis  of  some  German  experiments  by  Herr  Kollmann, 
which  is  reproduced  in  Table  XVII.  The  resistance  of  the 
materials  at  o°  Cent.,  or  32°  Fahr.,  is  taken  as  100,  and  that  at 
other  temperatures  as  the  proper  proportional  part  of  that 
number. 

It  will  be  noticed  that  these  German  experiments  show  a 
much  earlier  decrease  of  resistance  than  those  of  the  Franklin 
Institute. 

The  results  of  some  tests  of  a  grade  of  charcoal  boiler  plate 
at  three  different  temperatures  are  given  in  "  Effect  of  low  and 
Jiigh  temperatures  on  steel." 

Some  French  experiments  by  M.  Baudrimont  are  given  in 
the  "  Journal  of  the  Franklin  Institute  "  for  1850,  by  which  he 
found  that  at  the  temperatures  32°,  212°  and  392°  Fahr.,  iron 


Art.  32.] 


EFFECT  OF   TEMPERATURE. 


UNIVERSITY 


TABLE   XVII. 


TEMPERATURE. 

FIBROUS  IRON. 

FINE   GRAINED    IRON. 

BESSEMER  STEEL. 

Cent. 

Fahr. 

0° 

32° 

IOO 

IOO 

IOO 

IOO 

212 

IOO 

IOO 

IOO 

200 

392 

95 

IOO 

IOO 

300 

572 

90 

97 

94 

500 

932 

38 

44 

34 

700 

1,292 

16 

23 

18 

900 

1,652 

6 

12 

9 

1,000 

1,832 

4 

7 

7 

wire   gave   the    following   tensile   resistances,    in   pounds   per 
square  inch,  respectively  : 

291,510.00;         271,602.00;         298,620.00; 

These  resistances  are  most  extraordinarily  high,  but,  so  far 
as  the  influence  of  variation  of  temperature  is  concerned,  show 
nothing  discordant  with  the  preceding  results. 

The  same  experimenter  found  the  tensile  resistances  of 
gold,  platinum,  copper,  silver  and  palladium  to  decrease,  in 
every  instance,  as  the  temperature  increased  from  32°  to  392° 
FaJir. 

In  his  "  Useful  Information  for  Engineers,"  Second  Series, 
Sir  Wm.  Fairbairn  gives  the  results  of  numerous  experiments 
made  on  "  short "  specimens  of  plate  and  rivet  iron  at  different 
temperatures. 


250 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


TABLE    XVIII. 


TEMP.,   FAHR. 

BREAKING   WEIGHT    IN    POUNDS    PER 
SQUARE    INCH. 

STRESS   IN    REFERENCE   TO   FIBRE. 

o° 

49,009 

With. 

60 

40,357 

Across. 

60 

43,406 

Across. 

60 

With. 

no 

44,160 

Across. 

112 

42,088 

With. 

120 

40,625 

With. 

212 

39,935 

With. 

212 

45,68o 

Across. 

212 

49,500 

With. 

27O 

44,020 

With. 

340 

49,968 

With. 

340 

42,088 

Across. 

395 

46,086 

With. 

Scarcely  red. 

38,032 

Across. 

Dull  red. 

30,513 

Across. 

In  Table  XVIII.  will  be  found  the  results  of  his  experi- 
ments on  plate  iron.  On  the  whole,  the  table  would  seem  to 
show  a  point  of  greatest  resistance  at  about  270°  to  300°, 
though  so  many  irregularities  exist  that  little  or  no  law  can  be 
observed.  In  other  words  little  or  no  decrease  takes  place  at 
395°  or  below.  Much  diminution,  however,  is  seen  at  "  scarcely 
red  "  and  more  at  "  dull  red." 

Table  XIX.  shows  the  results  of  Fairbairn's  experiments  on 
rivet  iron  at  different  temperatures.  The  irregularities  are  less 
than  those  seen  in  Table  XVIII. ,  and  a  maximum  would  seem 
to  exist  at  about  325°. 

The  areas  of  the  normal  sections  of  the  plate  specimens 
varied  from  0.6  to  0.8  square  inch,  while  the  sectional  areas  of 
the  rivet-iron  specimens  were  about  0.2  or  0.25  square  inch. 

Other  results  for  wrought  iron  will  be  found  in  Table  IX. 
of  Art.  35. 


Art.  32.] 


EFFECT  OF   TEMPERATURE. 


251 


TABLE   XIX 


TEMPERATURE, 
FAHR. 

BREAKING  WEIGHT  IN    POUNDS 
PER   SQUARE    INCH. 

TEMPERATURE, 
FAHR. 

BREAKING  WEIGHT  IN   POUNDS 
PER   SQUARE   INCH. 

-30° 

63,239 

250 

82,174 

+  60 

61,971 

270 

83,098 

66 

63,661 

310 

80,570 

114 

70,845 

325 

87,522 

212 

82,676 

415 

8l,830 

212 

74,153 

435 

86,056 

212 

80,985 

Red  heal. 

36,076 

All  the  preceding  results,  while  irregular  to  some  extent, 
show  conclusively  that  no  essential  decrease  in  the  tensile  re- 
sistance of  wrought  iron  takes  place  below  about  500°  Fahr., 
while  a  possible  increase  at  that  temperature  may  exist  over 
that  at  any  below,  but  that  at  about  1,000°  it  may  lose  more 
than  a  half  of  its  resistance.  These  conclusions  are  of  the 
greatest  importance  in  the  construction  of  boilers. 


Effect  of  Low  Temperatures  on  Wrought  Iron. 

It  is  a  matter  of  common  observation  that  many  articles, 
large  and  small,  are  much  more  easily  broken  in  very  cold 
weather  than  at  higher  temperatures.  These  breakages  are 
undoubtedly  frequently  due  to  the  circumstances  in  which  the 
piece  broken  is  found  at  the  time  of  failure,  either  partly  or 
wholly. 

The  frozen,  and  consequently  less  yielding,  condition  of  the 


252  WROUGHT  IRON  IN  TENSION.  [Art.  32. 

ground  in  the  winter  is  unquestionably  a  very  potent  factor  in 
failures  or  tires  and  axles  of  railway  rolling  stock,  but  it  is  at 
least  an  open  question  whether  it  is  the  sole  cause. 

A  number  of  investigators  have  made  numerous  experi- 
ments with  the  object  of  determining  the  effect  of  low  tem- 
peratures on  the  resistance  of  wrought  iron  in  different  forms. 

From  the  results  of  these  experiments,  however,  they  have 
drawn  the  most  discordant  conclusions.  In  some  cases  this 
arises  from  the  fact  that  the  tests  have  not  been  made  under 
the  same  circumstances,  or  have  not  been  of  the  same  kind. 

Knut  Styffe  ("  Iron  and  Steel  ")  made  the  following  <l  Re- 
sume of  Results  of  Experiments  on  Tension  at  different  Tem- 
peratures :  "  , 

I.  "  That  the  absolute  strength  of  iron  and  steel  is  not  dimin- 
ished by  cold,  but  that  even  at  the  lowest  temperature 
which  ever  occurs  in  Sweden  it  is  at  least  as  great  as 
at  the  ordinary  temperature  (about  60°  Fahr^).  .  .  . 

3.  "  That  neither  in  steel  nor  in  iron  is  the  extensibility  less  in 

severe  cold  than  at  the  ordinary  temperature  ;    . 

4.  "  That  the  limit  of  elasticity  in  both  steel  and  iron  lies  higher 

in  severe  cold  ;     .     .     ." 

He  concluded  from  his  experiments  that  the  common  im- 
pression of  increased  weakness  and  brittleness  with  a  low  de- 
gree of  temperature  is  entirely  erroneous.  His  tests,  however, 
were  wholly  with  tension  gradually  applied,  and  could  support 
no  conclusion  in  regard  to  other  conditions. 

The  translator  of  Styffe's  work,  Christer  P.  Sandberg,  made 
some  experiments  in  order  to  determine  the  effect  of  shocks  at 
different  temperatures,  i.e.,  ordinary  and  low.  These  were  also 
made  in  Sweden,  and  by  dropping  heavy  weights,  from  differ- 
ent heights,  on  rails  supported  at  each  extremity.  The  records 
of  these  tests  may  be  found  in  the  translator's  Appendix  to 
Styffe's  work. 


Art.  32.]  EFFECT  OF   TEMPERATURE.  253 

The  following  are  Sandberg's  conclusions,  and  they  will  be 
observed  to  be  directly  opposed  to  those  of  Styffe  : 

1.  "  That  for  such  iron  as  is  usually  employed  for  rails  in  the 

three  principal  rail-making  countries  (Wales,  France 
and  Belgium),  the  breaking  strain,  as  tested  by  sudden 
blows  or  shocks,  is  considerably  influenced  by  cold ; 
such  iron  exhibiting  at  10°  Fahr.  only  one-third  to 
one-fourth  of  the  strength  which  it  possess  at  84° 
Fahr. 

2.  "  That  the  ductility  and  flexibility  of  such  iron  is  also  much 

affected  by  cold ;  rails  broken  at  10°  Fahr.  showing 
on  an  average  a  permanent  deflection  of  less  than  one 
inch,  whilst  the  other  halves  of  the  same  rails,  broken 
at  84°  Fahr.,  showed  a  set  of  more  than  four  inches 
before  fracture. 

3.  "  That  at  summer  heat  the  strength  of  the  Aberdare  rails 

was  20  per  cent,  greater  than  that  of  the  Creusot  rails ; 
but  that  in  winter  the  latter  were  30  per  cent,  stronger 
than  the  former." 

All  these  experiments  were  made  previous  to  1869,  and 
with  iron  rails. 

Prof.  Thurston,  from  his  own  experiments  and  those  of 
others,  concludes  (Trans.  Am.  Soc.  of  Civ.  Engrs.,  Vol.  III.,  p. 
30),  "  That  with  good  materials,  cold  does  not  produce  injury, 
but  actually  improves  their  power  of  resisting  stress  and  in- 
creases their  resilience. 

"  That  the  influence  of  impurities,  of  various  methods  of 
manufacture,  of  changes  of  density  with  temperature,  and  of 
the  causes  which  produce  a  concentration  of  the  action  of 
rapidly  produced  distortion  and  of  quick  blows,  are  subjects 
which  still  require  careful  investigation." 

He  considers  it  probable  that  the  cold-shortening  effect  of 
phosphorus  is  intensified  at  low  temperatures. 


254  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

After  observing  the  failures  on  the  railroads  coming  under 
their  observation,  the  Railroad  Commissioners  of  Massachu- 
setts reported  in  1874  that,  in  their  opinion,  neither  iron  nor 
steel  attained  any  greater  degree  of  brittleness,  or  became  any 
more  "  unreliable  for  mechanical  purposes "  at  low  tempera- 
tures than  at  ordinary.  They  did  not  observe  as  a  "  rule  that 
the  most  breakages  "  occurred  "  on  the  coldest  days." 

They  further  stated  that  "  the  introduction  of  steel  in  place 
of  iron  rails,  has  caused  an  almost  complete  cessation  of  the 
breakage  of  rails." 

Thus  it  is  seen  that  the  subject  is  most  thoroughly  involved 
in  confusion.  It  seems,  however,  to  be  established  that  the 
resistance  of  iron,  at  a  low  temperature,  to  a  steady  strain,  is 
not  diminished,  while  it  may,  perhaps,  be  increased. 

Its  resistance  to  shocks,  at  low  temperatures,  is  probably 
very  much  affected  by  its  quality,  mode  of  manufacture  or 
chemical  composition,  and  these  should  always  be  taken  into 
consideration  when  experiments  are  made. 

The  Report  of  the  Mass.  Railroad  Commissioners  would 
indicate  that  steel  rails  resist  shocks  at  low  temperatures  better 
than  iron  ones. 


Iron   Wire. 

Mr.  John  A.  Roebling  found  by  his  tests  that  the  English 
wire  used  in  the  Niagara  Falls  Suspension  Bridge  gave  an  ulti- 
mate tensile  resistance  of  about  98,500.00  pounds  per  square 
inch  ("  Papers  and  Practice  Illustrative  of  Public  Works."  John 
Weale,  London,  1856).  This  wire  was  about  0.145  inch  in 
diameter. 

The  Committee  of  the  Franklin  Institute  made  thirteen 
tests  of  some  iron  wire  one-third  of  an  inch  in  diameter,  of 
which  the  highest,  lowest  and  mean  ultimate  resistances  in 
pounds  per  square  inch  of  original  section  were  as  follows  : 


Art.  32.] 


IRON    WIRE. 


255 


Highest •  •  •; 88,354.00  pounds. 

Mean 84,186.00  pounds. 

Lowest 72,325.00  pounds. 

The  results  of  other  tests   by  the  same  committee  have 
already  been  given  under  "  Effect  of  annealing'' 


TABLE   XX. 


ULTIMATE   TENSILE   RESISTANCE   IN    POUNDS   PER 

ORIGINAL  DIAMETER 

SQUARE   INCH   OF 

CONTRACTION  OF 

IN   INCHES. 

ORIGINAL  AREA 

Original  Area. 

Fractured  Area. 

OF  SECTION. 

0.122 

94,871 

179,032 

0.47 

o.  123 

87,395 

162,500 

O.462 

O.I24 

89,256 

145,946 

0.388 

0.125 

88,618 

137,974 

0.358 

O.I22 

92,308 

168,750 

0-453 

o.  124 

9*,735 

156,338 

0.413 

0.124 

90,032 

170,313 

0.471 

O.  122 

92,308 

168,750 

0-453 

0.124 

91,735 

173,437 

0.471 

0.124 

86,776 

164,063 

0.471 

o.  125 

87,805 

156,522 

0-439 

o.  124 

86,776 

152,174 

o-43 

Table  XX.  is  a  condensed  form  of  one  given  in  the  4<  Trans- 
actions of  the  Am.  Soc.  Civ.  Engrs.,"  Vol.  III.,  p.  212,  and  con- 
tains an  account  of  the  tests  made  by  Prof.  R.  H.  Thurston  on 
some  wires  that  had  been  in  use  32  years  in  the  cables  of  the 
Fairmount  Suspension  Bridge  at  Philadelphia.  It  is  both  in- 
teresting and  important  to  observe  that  the  long  service  can- 
not have  appreciably  injured  either  the  ducility  or  ultimate 
resistance  of  the  wire. 

Table  XXI.  contains  the  records  of  tests  on  other  wire,  at 
the  same  time  (1875),  by  Prof.  Thurston.  The  small  reduction 


256 


WROUGHT  IRON  IN   TENSION. 


[Art.  32. 


of  diameter  at  fracture  shows  the  iron  to  have  been  not  very 
ductile.  It  will  also  be  noticed  that  the  smaller  diameters  give 
much  the  highest  resistances. 

TABLE   XXI. 


ORIGINAL  DIAMETER. 

DIAMETER   AFTER   FRACTURE. 

ULTIMATE  RESISTANCE  IN   POUNDS  PER 
SQUARE   INCH   OF   ORIGINAL  AREA. 

0.134 

0.133 

92,890 

0.1205 

O.II85 

84,442 

0.08 

0.0795 

94,299 

0.071 

0.068 

90,384 

0.0535 

O.O532 

105,871 

0.029 

0.029 

113,546 

According  to  Weisbach  ("  Mechanics  of  Engineering,  etc.," 
Vol.  I.,  4th  Edit.),  Lagerhjelm  and  Brix  found  the  mean  value 
of  the  -ultimate  resistance,  for  a  large  number  of  tests  of 
wrought-iron  wire  with  diameters  varying  from  0.0833  to  0.125 
inch,  to  be  98,000.00  pounds  per  square  inch  for  unannealed 
wire,  and  64,500.00  pounds  for  annealed. 

Morin,  in  his  "  Mecanique  Practique,"  gives  the  following 
for  unannealed  iron  wire,  after  changing  his  results  to  pounds 
per  square  inch  : 

Mean  for  diameters  of  0.039  *°  0.118  inch. . .  .  85,000.00  (nearly). 

Highest  for  diameters  of  0.02  to  0.039  inch. . .  114,000.00  (nearly). 

Lowest  for  large  diameter 71,000.00  (nearly). 

For  a  special  grade  ("  1'Aigle  ") 128,000.00  (nearly). 

SirWm.  Fairbairn  ("Useful  information  for  Engineers,  3d 
Series,"  p.  282)  gives  the  following  as  the  results  of  experi- 


Art.  32.] 


IRON    WIRE. 


257 


ments  on  various  kinds  of  English  iron  wire.  These  experi- 
ments resulted  from  investigations  relating  to  the  fabrication 
of  a  submarine  Atlantic  cable. 


KIND    OF    WIRE. 

DIAMETER. 

ULT.   RESIST. 

STRETCH. 

Haematite  

Inch. 
0.087 

Pounds. 
109  300 

Inch, 
o  280 

Homogeneous     

O.OQ"; 

I'll  OOO 

o  366 

Special  Homogeneous 

O.  OQ7 

115  ooo 

o  267 

Charcoal  .  .          .  .        

O.OQ'} 

no  400 

O    17^ 

Galvanized  

O.OQ8 

86  200 

o  108 

Homogeneous 

o  089 

IOJ.  ^OO 

o  190 

Homogeneous 

O.OQI 

192  2OO 

O    712 

Charcoal  

O.oqi 

92  2OO 

o  108 

Homogeneous 

o  088 

1  06  ooo 

•   vu 
o  218 

Charcoal 

O.OQ^ 

80  960 

o.  ^20 

Haematite    S    3  

0.089 

88  400 

o  171 

Haematite,  S.  4  

O.O95 

105,800 

o.  366 

Homogeneous 

O.  1  80 

AC  2OO 

o  480 

Homogeneous       

o.  148 

61  050 

o.  550 

Homogeneous  

O.OQ5 

134,000 

0.346 

Homogeneous 

O   OQ5 

77  600 

o.  116 

Special  Charcoal  

O.OQ5 

105,800 

o.  170 

The  ultimate  resistance  is  in  pounds  per  square  inch,  and 
the  stretch  is  the  total  amount  for  50  inches  of  length. 
Reviewing  the  values  given,  it  appears  : 

1.  That  wire  is  the  strongest  form  in  which  iron  can  be  used  to 

resist  tensile  stress  ; 

2.  That,  as  a  rule,  tffe  ultimate  tensile  resistance  increases  as  the 

diameter  of  the  wire  decreases. 

Tensile  Resistance  of  Shape  Iron. 

The  phenomena  exhibited  in  the  fracture  of  shape  iron  de- 
pend, to  a  great  extent,  on  the  character  of  the  piles  from 
which  it  is  rolled.    The  webs  of  ES  and  Is  are  sometimes  rolled 
from  old  rails  in  connection  with  double  refined  iron  in  the 
17 


258  WROUGHT  IRON  IN  TENSION.  [Art.  32. 

flanges.  In  such  cases,  specimens  cut  from  the  web  will  fre- 
quently, if  not  usually,  show  a  high  intensity  of  ultimate  resist- 
ance, but  very  little  ductility,  while  those  cut  from  the  flanges 
will  give  good  records  of  both  kinds. 

In  general,  shapes  will  offer  less  tensile  resistance  than 
either  bars  or  rods,  yet  small  specimens  cut  from  good  shape 
iron  will  give  values  ranging  from  52,000  to  58,000  pounds  per 
square  inch,  with  ductility  little  less  than  that  of  Qs  and  Qs. 

English   Wrought  Iron. 

A  great  number  of  experiments  on  English  wrought  iron 
have  been  made  by  Sir  Wm.  Fairbairn,  David  Kirkaldy,  and 
others.  A  record  of  Fairbairn's  experiments  may  be  found  in 
his  "  Useful  Information  for  Engineers,"  while  an  account  of 
those  of  the  latter  is  given  in  "  Experiments  on  Wrought  Iron 
and  Steel,"  by  David  Kirkaldy,  Glasgow,  1863. 

B.  B.  Stoney,  in  his  "  Theory  of  Strains  in  Girders  and  Simi- 
lar Structures,"  summarizes  Kirkaldy's  results,  in  pounds  per 
square  inch,  as  follows : 

Mean  of  188  rolled  bars 57,555-QO 

Mean  of    72  angle  irons  and  straps 54, 729.00 

Mean  of  167  plates,  lengthwise 5°,737-OO 

Mean  of  160  plates,  crosswise 46,171.00 

It  should  be  stated  that  these  means  iftclude  some  Russian 
and  Swedish  irons,  also  that  the  bars  were  small  ones. 

These  results  do  not  differ  much  from  quantities  for  cor- 
responding grades  of  American  iron. 


Fracture  of  Wrought  Iron. 

The  characteristic  fracture  of  wrought  iron  broken  in  ten- 
sion, either  directly  or  transversely,  is  rather  coarsely  fibrous, 


Art.  32.]  FRACTURE  AND   CRYSTALLIZATION.  259 

not  unfrequently  exhibiting  a  few  bright  granular  spots  which, 
in  rare  cases,  may  possibly  be  crystalline.  This  characteristic 
(fibrous)  fracture  is  always  produced  by  the  steady  application 
of  an  external  force,  under  the  influence  of  which  the  piece  is 
drawn  out  in  jagged  points  at  the  place  of  failure. 

best  of  fibrous   wrought   iron,  however,  will  exhibit  a 


granular  fracture  if  broken  suddenly.  In  making  tests,  there- 
fore, it  is  of  the  greatest  importance  to  observe  and  direct  the 
mode  of  application  of  the  external  forces  producing  fracture. 

When  some  grades  of  iron  in  bars  are  broken  transversely 
by  shocks  (such  as  are  produced  by  falling  weights),  a  phenome- 
non known  as  "  barking  "  is  produced.  A  skin  of  metal  from 
a  sixteenth  to  an  eighth  of  an  inch  in  thickness,  on  the  tension 
side  of  the  bent  piece,  tears  apart  and  separates  from  the  core 
of  the  bar.  At  the  place  of  fracture  and  on  each  side  of  it, 
this  skin  or  "  bark  "  remains  essentially  straight.  This  kind  of 
fracture  shows  remarkably  well  the  fibrous  character  of  wrought 
iron  ;  it  is  simply  the  separation  of  the  fibres  near  the  outside 
of  the  bar  from  those  within. 

^ 

Crystallization  of  Wrought  Iron. 

The  subject  of  crystallization  of  wrought  iron  is  one  about 
which  there  is  much  dispute.  In  "  Strength  of  Wrought  Iron 
and  Chain  Cables,"  by  Beardslee,  as  abridged  by  Kent,  p.  36, 
the  following  is  given  as  the  opinion  or  view  of  the  United 
States  Testing  Commission  :  "  The  question  as  to  whether 
crystallization  can  be  produced  in  iron  by  stress,  or  by  repeti- 
tion of  stress  with  alternations  of  rest,  or  by  vibration,  has 
been  much  discussed,  and  very  opposite  views  are  entertained 
by  experts. 

"  We  have  met  with  but  one  unmistakable  instance  of  crystal- 
lization which  was  probably  produced  by  alternations  of  severe 
stress,  sudden  strains,  recoils  and  rest." 

The   committee  then   state  the  case  of  a  connecting-rod, 


260  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

carefully  made  of  the  best  quality  of  wrought-iron  scrap,  which 
had  been  used  in  a  testing  machine  for  forty  years,  in  the 
Navy  Yard  at  Washington.  It  was  five  inches  in  diameter, 
but  one  day,  while  in  use  it  suddenly  broke  under  a  stress 
(total)  of  less  than  200,000  pounds.  "  The  surface  of  the  fract- 
ured ends  showed  well-defined  crystallization,  the  facets  being 
large  and  bright  as  mica." 

The  data  at  hand,  at  present,  are  not  sufficient  for  a  decision 
of  the  question,  but  it  may  be  confidently  stated  that  in  many 
cases  granulation  has  been  mistaken  for  crystallization. 


Elevation  of  Ultimate  Resistance  and  Elastic  Limit. 

It  was  first  observed  by  Prof.  R.  H.  Thurston  and  Com- 
mander L.  A.  Beardslee,  U.  S.  N.,  independently,  in  this  coun- 
try, that  if  wrought  iron  be  subjected  to  a  stress  beyond  its 
elastic  limit,  but  not  beyond  its  ultimate  resistance,  and  then 
allowed  to  "  rest  "  for  a  definite  interval  of  time,  a  considerable 
increase  of  elastic  limit  and  ultimate  resistance  may  be  expe- 
rienced. In  other  words,  the  application  of  stress  and  subse- 
quent "  rest  "  increases  the  resistance  of  wrought  iron. 

This  "  rest  "  may  be  an  entire  release  from  stress  or  a  sim- 
ple holding  the  test  piece  at  a  given  intensity. 

Prof.  Thurston's  investigations  were  on  torsion,  while  those 
of  the  United  States  Commission  were  on  tension,  and  will  be 
given  here. 

The  Commission  prepared  twelve  specimens  and  subjected 
them  to  an  intensity  of  stress  equal  to  the  ultimate  resistance 
of  the  material,  without  breaking  the  specimens.  These  were 
then  allowed  to  rest,  entirely  free  from  stress,  from  twenty-four 
to  thirty  hours,  after  which  period  they  were  again  stressed 
until  broken. 

The  gain  in  ultimate  resistance  by  the  rest  was  found  to 
vary  from  4.4  to  17  per  cent. 


Art.  32.]  ELEVATION  OF  RESISTANCE.  26 1 

These  tests,  remark  the  committee,  seem  to  indicate  that 
the  tough  fibrous  irons  gained  the  most,  while  those  which 
broke  with  a  steel-like  fracture  gained  the  least. 

Before  the  rest,  the  stress  which  produced  the  first  perma- 
nent elongation  was  about  65  per  cent,  of  the  ultimate  resist- 
ance, but  after  the  rest  the  two  were  nearly  identical. 

The  ^committee  then  took  forty-two  other  specimens  and 
subjected  them  to  precisely  the  same  operations,  except  that 
the  rest  periods  varied  from  one  minute  to  six  months. 

The  gains  were  as  follows : 

In  less  than  I  hour 1. 1  per  cent.,  mean  of    5  tests. 

In  less  than  8  and  over  I  hour 3.8  per  cent.,  mean  of    8  tests. 

In  3  days 16.2  per  cent.,  mean  of  10  tests. 

In  8  days 17.8  per  cent.,  mean  of    2  tests. 

Between  8  and  43  days 15.3  per  cent. ,  mean  of    5  tests. 

In  6  months 17.9  per  cent.,  mean  of  12  tests. 

After  seven  other  experiments  involving  a  rest  of  24  hours, 
with  an  average  gain  of  15.4  per  cent.,  the  committee  con- 
cluded "  that  at  the  end  of  one  day  the  result  is,  with  very 
ductile  irons,  practically  accomplished." 

The  manifestation  of  this  phenomenon  in  different  grades 
of  iron  was  then  investigated. 

"  Thirteen  pieces  were  prepared,  five  of  which  were  of  soft 
charcoal  bloom  boiler  iron,  five  of  coarse  contract  chain  iron, 
and  three  of  a  fine-grained  bar  of  ...  very  pure  iron  with 
high  tenacity." 

After  testing  these  specimens  subsequent  to  an  eighteen 
hours'  rest,  the  committee  state  (Kent's  abridgment) : 

"These  experiments  confirmed  the  opinion  already  formed, 
and  indicate  that  a  bridge,  cable,  or  other1  structure,  composed 
of  iron  of  either  of  the  latter  two  varieties,  will  receive  com- 
paratively slight  benefit  from  the  operation  of  this  law ;  while 
ductile  fibrous  metal  .  .  .  gains  .  .  .  to  a  great  extent 
by  the  effect  of  strains  already  withstood."  The  gain  in  these 


262  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

specimens  varied  from  about  3  per  cent,  (for  the  coarse  iron) 
to  about  1 8  per  cent,  (for  the  soft  iron). 

Again,  two  sets  of  specimens  were  prepared :  one  from  the 
two  portions  of  fractured  bars  after  having  been  pulled  asunder, 
the  other  from  the  bars  in  their  normal  condition.  After  a  rest 
of  several  days  the  first  set  showed  a  gain  over  the  second  in 
ultimate  resistance,  varying  from  about  8  to  39  per  cent.,  the 
higher  values  belonging  to  the  more  ductile  irons. 

B auscl Linger  s  Experiments  on  the  Change  of  Elastic  Limit  and 
Coefficient  of  Elasticity. 

In  "  Der  Civilingenieur,"  Heft  5,  for  1881,  are  contained  the 
results  of  the  experiments  of  Prof.  Bauschinger,  of  Munich. 
The  observations  in  these  experiments  were  made  by  the  aid 
of  a  piece  of  apparatus  which  gave  the  elongations  (all  experi- 
ments were  tensile)  in  ten-millionths  of  a  metre,  or  approxi- 
mately in  sjnjVrro-  °^  an  inc^*  ^n  extraordinarily  high  degree 
of  accuracy  was  therefore  attained. 

Prof.  Bauschinger's  elastic  limit  was  strictly  a  proportion- 
ality limit  between  stresses  and  strains.  He  also  observed 
what  may  be  called  the  "stretch-limit  "  (Ger.,  Streckgrenze),  at 
which  point  the  stretching  or  elongation  suddenly  increases  and 
continues  to  increase  for  more  than  a  minute  after  the  appli- 
cation of  the  stress.  In  ordinary  experimenting  this  point  has 
probably  frequently  been  considered  the  elastic  limit. 

The  test  pieces  were  subjected  to  loads  which  gradually 
increased  from  zero  by  an  increment  a  little  less  than  3,000 
pounds  per  square  inch,  each  load  having  been  allowed  to  act 
one  minute  before  adding  the  succeeding  increment.  At  inter- 
vals of  the  loading  separated  by  about  11,500  or  12,000  pounds 
per  square  inch,  each  piece  was  entirely  unloaded  and  allowed 
to  remain  so  for  15  or  20  minutes.  After  the  "  stretch-limit" 
was  found  the  piece  was  subjected  to  a  final  load  somewhat 
greater  than  the  "  stretch-limit/'  and  then  entirely  unloaded. 


Art.  32.] 


BAUSCHINGEFS  EXPERIMENTS. 


263 


In  some  cases  the  piece  was  immediately  put  through  the 
same  process  of  testing  either  once  or  a  number  of  times,  and 
the  results  of  such  tests  will  be  found  in  the  columns  of  the 
following  tables,  indicated  by  the  contraction  "  Imy" 

In  the  remaining  cases  intervals  of  time,  shown  at  the  tops 
of  the  columns,  were  allowed  to  elapse  between  any  one  test 
and  the  succeeding  one. 

The  tables,  Nos.  I  to  7  inclusive,  give  the  results  of  the 
experiments  on  seven  specimens  of  a  grade  of  iron  called 
"  Schweisseisen "  (weld  iron).  These  specimens  were  a  very 
little  less  than  I  inch  (25  millimetres)  in  diameter.  Nos.  I  and 
2  were  about  32  inches  long,  and  the  others  about  16  inches 
long. 

Tables  No.  8  to  13,  inclusive,  give  the  results  obtained  with 
Krupps  "  Flusseisen."  These  specimens  were  about  one  inch 
in  diameter  and  sixteen  inches  long. 

The  tables  have  been  condensed  from  those  given  by  Bau- 
schinger  and  reduced  to  English  measures. 

The  following  is  the  notation  : 


E.  L. 
S.-L. 
R  L. 
E. 


elastic  limit  in  pounds  per  square  inch, 
stretch  limit  in  pounds  per  square  inch, 
final  load  in  pounds  per  square  inch, 
coefficient  of  elasticity  in  pounds  per  sq.  in. 


Weld  Iron. 


NO.     I. 

IN    ORIGINAL    CONDI- 
TION. 

IM'Y. 

IM'Y. 

IM'Y. 

E.  L. 

20,110 

14,370 

14,900 

15,500 

S.-L. 

27,300 

31,600 

41,700 

49,500 

F.  L. 

3I,600 

4O,2OO 

47,700 



E. 

39,293,000 

27,928,000 

27,672,000 

27,544,000 

264 


WROUGHT  IRON  IN   TENSION.  [Art.  32. 


Weld  Iron. 


NO.    2. 

IN    ORIGINAL    CONDI- 
TION. 

AFTER  19  MRS. 

AFTER  27  HRS. 

AFTER  24  HRS. 

E.  L. 

20,  no 

28,970 

35,500 

39,100 

S.-L. 

28,700 

34,750 

44,300 

45,ioo 

F.  L. 

31,600 

40,540 

47,300 

— 

E. 

29,037,000 

28,923,OOO 

28,198,000 

28,241,000 

Weld  Iron. 


IN    ORIGINAL    CONDI- 

NO.   3. 

TION. 

AFTER   51  HRS. 

AFTER   41  HRS. 

AFTER  45  HRS. 

E.  L. 

23,164 

28,44O 

39,080 

45,530 

S.-L. 

29,OOO 

34,750 

45,090 



F.  L. 

31,900 

40,540 

48,100 



E. 

29,208,000 

28,397,000 

28,483,000 

28,170,000 

Weld  Iron. 


IN    ORIGINAL    CONDI- 

NO.   4. 

AFTER   80  HRS. 

AFTER  68  HRS. 

AFTER   64  HRS. 

TION. 

E.  L. 

22,890 

31,900 

35,340 

43,800 

S.-L. 

30,050 

34,750 

44,170 

F.  L. 

31,470 

40,540 

47,no 



E. 

29,293,OOO 

28,810,000 

28,227,000 

28,696,000 

Art.  32.]  BAUSCHINGER'S  EXPERIMENTS. 


265 


Weld  Iron. 


IN    ORIGINAL    CONDI- 

NO.     5. 

IM'Y. 

AFTER  63  HRS. 

IM'Y. 

E.  L 

21,070 

14,720 

42,090 

15,260 

S.-L. 

3O,67O 

35,320 

48,110 

51,860 

F.  L. 

34,6n 

42,700 

51,110 



E. 

29,293,000 

28,312,000 

28,056,000 

26,705,000 

Weld  Iron. 


IN    ORIGINAL    CONDI- 

NO.    6. 

TION. 

AFTER  48.5  HRS. 

AFTER  44.5  HRS. 

AFTER   49  HRS. 

E.  L. 

27,730 

26,720 

33,350 

24,940 

S.-L. 

32,120 

37,no 

45,480 



F.  L. 

35,040 

43,040 

51,550 



E. 

29,720,000 

28,639,000 

28,483,OOO 

28,881,000 

Weld  Iron. 


IN    ORIGINAL    CONDI- 

NO.   7. 

TION. 

AFTER   47  HRS. 

AFTER  50.5  HRS. 

AFTER  42.5  HRS. 

E.  L. 

20,110 

26,720 

27,170 

18,540 

S.-L. 

30,160 

38,590 

45,290 



F.  L. 

34,470 

43,040 

51,320 



E. 

28,668,000 

28,6lI,OOO 

28,568,000 

29,592,000 

266  WROUGHT  IRON  IN   TENSION.  [Art.  32, 


Melted  Wrought  Iron. 


NO.    8. 

IN    ORIGINAL    CONDI- 
TION. 

IM'Y. 

IM'Y. 

IM'Y. 

E.  L. 

35,340 



8,990 

9,230 

S.-L. 

36,750 

46,910 

53,890 

58,490 

F.  L. 

45,230 

52,770 

59,880 



E. 

3I,256,OOO 

31,483,000 

30,488,000 

Melted  Wrought  Iron. 


IN    ORIGINAL    CONDI- 

NO.    9. 

IM'Y. 

IM'Y. 

IM'Y. 

TION. 

E.  L. 

37,870 

5,770 

14,720 

I5,l6o 

S.-L. 

42,080 

46,  140 

53,000 

60,630 

F.  L. 

44,880 

5I,920 

58,880 



E. 

32,379,000 

31,796,000 

29,947,000 

28,2I3,OOO 

Melted  Wrought  Iron. 


NO.  10. 

IN    ORIGINAL    CONDI- 
TION. 

AFTER  3  HRS. 

AFTER    15  HRS. 

AFTER  7  HRS. 

E.  L. 

33,790 

11,490 

17,730 

15,260 

S.-L. 

36,600 

43,090 

53,2io. 

6l,O2O 

F.  L. 

42,230 

51,704 

69^3° 



E. 

3I,88l,OOO 

31,953,000 

31,895,000 

32,393,000 

Art.  32.]  BAUSCHINGER'S  EXPERIMENTS. 


267 


Melted  Wrought  Iron. 


IN    ORIGINAL    CONDI- 

NO. II. 

TION. 

AFTER  2.5  HRS. 

AFTER  15.5  HRS. 

AFTER  5.5  HRS. 

E.  L. 

33,930 

11,590 

n,774 

I2,O7O 

S.-L. 

39,570 

43,442 

53,ooo 

60,380 

F.  L. 

42,404 

52,130 

58,880 



E. 

32,536,000 

32,237,000 

32,222,000 

31,085,000 

Melted  Wrought  Iron. 


IN    ORIGINAL    CONDI- 

NO.  12. 

TION. 

AFTER    51  HRS. 

AFTER   47  HRS. 

AFTER   46  HRS. 

E.  L. 

36,900 

43,800 

39,82O 

40,950 

S.-L. 

39,731           • 

49,640 

55,940 

60,630 

F.  L. 

42,630 

52,560 

58,880 



E. 

32,479,000 

31,754,000 

31,696,000 

31,568,000 

Melted  Wrought  Iron. 


NO.  13. 

IN   ORIGINAL 
CONDITION. 

AFTER 
43-5    HRS. 

AFTER 
54    HRS. 

AFTER 
44-5   HRS. 

AFTER 
45-5    HRS. 

AFTER 
10    DAYS. 

E.  L. 



35,340 

33,930 

41,740 

42,720 

6l,560 

S.-L. 



36,745 

47,600 

56,640 





F.  L. 



42,400 

51,920 

59,630 





E. 

31,853,000 

32,165,000 

31,298,000 

31,454,000 

31,440,000 

32,364,000 

268  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

During  the  progress  of  the  various  tests,  the  bars  Nos.  6,  7, 
9,  ii  and  12  were  subjected  to  shocks  in  addition  to  the  static 
tests.  These  shocks  were  produced  by  striking  the  test  piece 
on  its  end  by  a  hammer.  It  does  not  appear  that  these  blows 
of  the  hammer  perceptibly  influenced  the  results. 

The  ultimate  resistance  of  the  weld  iron  was  found  to  vary 
from  55,300  to  58,870  pounds  per  square  inch.  That  of  the 
melted  wrought  iron  was  about  65,000  pounds  per  square  inch. 

Although  there  are  some  irregularities,  the  following  gen- 
eral conclusions  may  be  drawn  from  the  tables: 

By  "  immediate  "  testing  the  elastic  limit  of  weld  iron  is 
very  much  decreased. 

With  a  rest  (entirely  free  from  load)  between  the  tests,  the 
elastic  limit  of  weld  iron  is  very  much  increased. 

The  greatest  proportional  gain,  except  in  the  case  of  previ- 
ous immediate  testing,  seems  to  be  acquired  after  a  rest  no 
greater  than  twenty  hours. 

Bar  No.  6  is  seen  to  give  anomalous  results. 

In  all  cases  of  the  weld  iron  the  stretch-limit  is  considerably 
raised  by  repeated  testing. 

In  no  case  is  the  coefficient  of  elasticity,  after  once  testing, 
equal  to  its  original  value ;  as  a  rule,  a  steady  decrease  is  seen 
to  take  place  by  repeated  testing,  but  there  are  some  ex- 
ceptions. 

The  elastic  limit  of  "  Flusseisen,"  after  repeated  testing,  is 
found  to  be  much  less  than  its  original  value  until  the  length 
of  rest  becomes  about  fifty  hours. 

The  stretch-limit  of  the  same  metal  is  invariably  raised  by 
repeated  testing,  either  with  or  without  "  rests." 

In  nearly  all  the  cases  of  Nos.  8  to  13,  the  coefficient  of 
elasticity  is  found  to  be  slightly  decreased  by  repeated  testing. 

For  a  very  clear  and  detailed  account  of  these  experiments 
reference  must  be  made  to  the  "  Civilingenieur." 


Art.  32.]  SUDDEN  STRESS.  '269 


Resistance  of  Bar  Iron  to  Suddenly  Applied  Stress. 

If  tensile  stress  is  suddenly  applied  to  a  bar  of  wrought  iron,       **/ 
both  its  ultimate  resistance  and  elongation  will  be  very  materi-  \ 
ally  decreased. 

As  a  mean  of  a  number  of  tests,  Mr.  David  Kirkaldy 
(u  Experiments  on  Wrought  Iron  and  Steel  ")  found  with  sud- 
denly applied  stress  an  ultimate  resistance  of  46,500  pounds 
per  square  inch,  while  with  stress  gradually  applied  it  rose  to 
57,200  pounds. 

In  the  former  case  the  elongation  was  about  20  per  cent., 
and  as  high  as  24.6  per  cent,  in  the  latter. 

It  is  thus  seen  that  the  mode  of  application  of  external 
force  not  only  affects  the  character  of  the  fracture  of  the  iron, 
but  also  its  ultimate  resistance  and  elongation. 

It  will  hereafter  be  seen  that  similar  observations  apply  to 
other  metals  than  wrought  iron. 


Reduction   of  Resistance  Between   the   Ultimate  and  Breaking 

Point. 

It  has  already  been  observed  that  the  ultimate  tensile  re- 
sistance of  wrought  iron  is  the  greatest  tensile  resistance  which 
it  offers  to  being  pulled  asunder,  and  that  a  test  specimen 
finally  parts  at  much  less  than  the  ultimate  resistance.  This  is 
due  to  the  ductility  of  the  iron,  which  allows  it  to  "  pull  out  " 
or  stretch,  thus  decreasing  the  cross  section  as  well  as  the 
actual  resisting  capacity  of  the  metal. 

The  ultimate  resistance,  therefore,  is  not  exerted  on  the  final 
section  of  fracture,  but  on  a  section  somewhat  greater  ;  referring 
it  (the  ultimate  resistance)  to  the  section  of  fracture,  then,  may 
mean  little  or  nothing. 

The  United  States  Commission  made  six  tests,  for  the  pur- 
pose of  determining  this  reduction,  on  some  specimens  which 


2/0  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

had   previously  been   stressed   with   a  subsequent  rest.     The 
highest,  lowest,  and  mean  losses  were  as  follows  : 

Highest 14- 5  per  cent. 

Mean 13  •  8  per  cent. 

Lowest 12.9  per  cent. 

It  was  observed  from  a  number  of  specimens,  by  the  same 
commission,  that  the  reduction  of  area  at  the  instant  of  ulti- 
mate resistance  (or  greatest  resistance)  was  about  one-half,  and 
the  elongation  or  strain  a  little  over  three-quarters,  of  the  cor- 
responding quantities  at  the  instant  of  fracture,  supposing 
failure  to  be  produced  by  a  steady  strain. 

Some  further  observations  seemed  to  show  that  if  failure 
were  produced  by  shock,  the  final  contraction 'would  be  nearly 
the  same  as  at  the  instant  of  greatest  resistance  in  the  case  of 
a  steady  failure. 


Effects  of  Chemical  Constitution. 

While  it  is  well  known  that  the  resistance  of  wrought  iron 
to  tension  varies  greatly  with  the  chemical  composition,  it  is 
yet  uncertain  just  what  influence  most  of  the  foreign  elements, 
found  in  iron  exert,  either  individually  or  collectively.  This 
will  be  apparent  on  examining  Table  XXII.,  taken  from  the 
report  of  the  three  committees  of  the  United  States  Commis- 
sion, to  which  allusion  has  here  been  so  frequently  made 
before. 

The  first  part  of  the  table  represents  the  relative  values  of 
sixteen  different  irons  in  reference  to  their  physical  character- 
istics, one  being  the  highest.  The  second  part  shows  the 
amount  of  the  various  elements  named  in  the  left-hand  lower 
column,  found  in  the  corresponding  irons,  i.  e.,  each  vertical 
column  belongs  to  one  iron. 

An  inspection  of  the  table  will  make  very  evident  the  diffi- 


Art.  32.] 


CHE  MIC  A  L    CONS  TI TU  TION. 


271 


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NUMBER  — 

rt                  o 

£>    .2      ~      ^ 

11        l| 

C        'O          O          0) 

—        —        —        — 

In  power  of  resisting  shock. 

£• 

CM 

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2/2  WROUGHT  IRON  IN   TENSION.  [Art.  32. 

culty  of  drawing   definite  conclusions  in  regard   to  any  one 
element. 

For  a  detailed  discussion  of  these  results  reference  must  be 
made  to  the  report. 

Kirkaldys  Conclusions. 

The  following  conclusions  were  deduced  by  Mr.  Kirkaldy 
from  the  results  of  his  experiments.  As  will  be  seen,  they 
belong  to  both  wrought  iron  and  steel  in  tension,  and  are 
taken  from  his  "  Experiments  on  Wrought  Iron  and  Steel," 
1861  : 

1.  The  breaking  strain  does  not  indicate  the  quality,  as  hitherto  assumed. 

2.  A  high  breaking  strain  may  be  due  to  the  iron  being  of  superior  quality, 
dense,  fine,  and  moderately  soft,  or  simply  to  its  being  very  hard  and  unyielding. 

3.  A  low  breaking  strain  may  be  due  to  looseness  and  coarseness  in  the  texture, 
or  to  extreme  softness,  although  very  close  and  fine  in  quality. 

4.  The  contraction  of  area  at  fracture,  previously  overlooked,  forms  an  essential 
element  in  estimating  the  quality  of  specimens. 

5.  The  respective  merits  of  various  specimens  can  be  correctly  ascertained  by 
comparing  the  breaking  strain  jointly  with  the  contraction  of  area. 

6.  Inferior  qualities  show  a  much  greater  variation  in  the  breaking  strain  than 
superior. 

7.  Greater  differences  exist  between  small  and  large  bars  in  coarse  than  in  fine 
varieties. 

8.  The  prevailing  opinion  of  a  rough  bar  being  stronger  than  a  turned  one  is 
erroneous. 

9.  Rolled  bars  are  slightly  hardened  by  being  forged  down. 

10.  The  breaking  strain  and  contraction  of  area  of  iron  plates  are  greater  in  the 
direction  in  which  .they  are  rolled  than  in  a  transverse  direction. 

11.  A  very  slight  difference  exists  between  specimens  from  the  centre  and  speci- 
mens from  the  outside  of  crank  shafts. 

12.  The  breaking  strain  and  contraction  of  area  are  greater  in  those  specimens 
cut  lengthways  out  of  crank  shafts  than  in  those  cut  crossways. 

13.  The  breaking  strain  of  steel,  when  taken  alone,  gives  no  clue  to  the  real 
qualities  of  various  kinds  of  that  metal. 

14.  The  contraction  of  area  at  fracture  of  specimens  of  steel  must  be  ascertained 
as  well  as  in  those  of  iron. 

15.  The  breaking  strain,  jointly  with  the  contraction  of  area,  affords  the  means 
of  comparing  the  peculiarities  in  various  lots  of  specimens. 


Art.  32.]  KIRKALDY'S  CONCLUSIONS.  273 

16.  Some  descriptions  of  steel  are  found  to  be  very  hard,  and,  consequently, 
suitable  for  some  purposes  ;  whilst  others  are  extremely  soft,  and  equally  suitable 
for  other  uses. 

17.  The  breaking  strain  and  contraction  of  area  of  puddled  steel  plates,  as  in  iron 
plates,  are  greater  in  the  direction  in  which  they  are  rolled  ;  whereas  in  cast  steel 
they  are  less. 

1 8.  Iron,  when  fractured  suddenly,  presents  invariably  a  crystalline  appearance ; 
when  fractured  slowly,  its  appearance  is  invariably  fibrous. 

19.  The  appearance  may  be  changed  from  fibrous  to  crystalline  by  merely  al- 
tering the  shape  of  specimen,  so  as  to  render  it  more  liable  to  snap. 

20.  The  appearance  may  be  changed  by  varying  the  treatment,  so  as  to  render 
the  iron  harder  and  more  liable  to  snap. 

21.  The  appearance  may  be  changed  by  applying  the  strain  so  suddenly  as  to 
render  the  specimen  more  liable  to  snap,  from  having  less  time  to  stretch. 

22.  Iron  is  less  liable  to  snap  the  more  it  is  worked  and  rolled. 

23.  The  "  skin  "  or  outer  part  of  the   iron  is  somewhat  harder  than  the  inner 
part,  as  shown  by  appearance  of  fracture  in  rough  and  turned  bars. 

24.  The  mixed  character  of  the  scrap  iron  used  in  large  forgings  is  proved  by 
the  singularly  varied  appearance   of  the    fractures  of  specimens  cut  out  of  crank 
shafts. 

25.  The  texture  of  various  kinds  of  wrought  iron  is  beautifully  developed  by 
immersion  in  dilute  hydrochloric  acid,  which,  acting  on  the  surrounding  impurities, 
exposes  the  metallic  portion  alone  for  examination. 

26.  In  the  fibrous  fractures  the  threads  are  drawn  out,  and  are  viewed  externally, 
whilst  in  the  crystalline  fractures  the  threads  are  snapped  across  in  clusters,  and  are 
viewed  internally  or  sectionally.      In  the  latter  cases  the  fracture  of  the  specimen  is 
always  at  right  angles  to  the  length  ;  in  the  former  it  is  more  or  less  irregular. 

27.  Steel  invariably  presents,  when  fractured  slowly,  a  silky  fibrous  appearance  ; 
when  fractured  suddenly,  the  appearance  is  invariably  granular,  in  which  case  also 
the  fracture  is  always  at  right  angles  to  the  length  ;  •when  the  fracture  is  fibrous,  the 
angle  diverges  always  more  or  less  from  90°. 

28.  The  granular  appearance  presented  by  steel  suddenly   fractured  is  nearly 
free  of  lustre,  and  unlike  the  brilliant  crystalline  appearance  of  iron  suddenly  fract- 
ured ;  the  two  combined  in  the  same  specimen  are  shown  in  iron  bolts  partly  con- 
verted into  steel. 

29.  Steel  which  previously  broke  with  a  silky  fibrous  appearance,  is  changed 
into  granular  by  being  hardened. 

30.  The  little  additional  time  required  in  testing  those  specimens,  whose  rate  of 
elongation  was  noted,  had  no  injurious  effect  in  lessening  the  amount  of  breaking 
strain,  as  imagined  by  some.    • 

31.  The  rate  of  elongation  varies  not  only  extremely  in  different  qualities,  but 
also  to  a  considerable  extent  in  specimens  of  the  same  brand. 

32.  The  specimens  were  generally  found  to  stretch  equally  throughout  their 

18 


2/4  WROUGHT  IRON  IN   TENSION,  [Art.  32. 

length  until  close  upon  rupture,  when  they  more  or  less  suddenly  drew  out,  usually 
at  one  part  only,  sometimes  at  two,  and,  in  a  few  exceptional  cases,  at  three  differ- 
ent places. 

33.  The  ratio  of  ultimate  elongation  may  be  greater  in  short  than  in  long  bars 
in  some  descriptions  of  iron,  whilst  in  others  the  ratio  is  not  affected  by  difference 
in  the  length. 

34.  The  lateral  dimensions  of  specimens  forms  an  important  element  in  com- 
paring either  the  rate  of,  or  the  ultimate,  elongation — a  circumstance  which  has 
been  hitherto  overlooked. 

35.  Steel  is  reduced  in  strength  by  being  hardened  in  water,  while  the  strength 
is  vastly  increased  by  being  hardened  in  oil. 

36.  The  higher  steel  is  heated  (without,  of  course,  running  the  risk  of  being 
burned)  the  greater  is  the  increase  of  strength  by  being  plunged  into  oil. 

37.  In  a  highly  converted  or  hard  steel  the  increase  in  strength  and  in  hardness 
is  greater  than  in  a  less  converted  or  soft  steel. 

38.  Heated  steel,  by  being  plunged  into  oil  instead  of  water  is  not  only  consid- 
erably hardened^  but  toughened  by  the  treatment. 

39.  Steel  plates  hardened  in  oil,  and  joined  together  with  rivets,  are  fully  equal 
in  strength  to  an  unjointed  soft  plate,  or  the  loss  of  strength  by  riveting  is  more 
than  counterbalanced  by  the  increase  in  strength  by  hardening  in  oil. 

40.  Steel  rivets,  fully  larger  in  diameter  than  those  used  in  riveting  iron  plates 
of  the  same  thickness,  being  found  to  be  greatly  too  small  for  riveting  steel  plates, 
the  probability  is  suggested  that  the  proper  proportion  for  iron  rivets  is  not,  as 
generally  assumed,  a  diameter  equal  to  the  thickness  of  the  two  plates  to  be  joined. 

41.  The  shearing  strain  of  steel  rivets  is  found  to  be  about  a  fourth  less  than 
the  tensile  strain. 

42.  Iron  bolts,  case-hardened,  bore  a  less  breaking  strain   than  when  wholly 
iron,  owing  to  the  superior  tenacity  of  the  small  proportion  of  steel  being  more  than 
counterbalanced  by  the  greater  ductility  of  the  remaining  portion  of  iron. 

43.  Iron  highly  heated  and  suddenly  cooled  in  water  is  hardened,  and  the  break- 
ing strain,  when  gradually  applied,  increased,  but  at  the  same  time  it  is  rendered 
more  liable  to  snap. 

44.  Iron,  like    steel,   is  softened,   and  the  breaking   strain  reduced,  by  being 
heated  and  allowed  to  cool  slowly. 

45.  Iron  subject  to  the  cold-rolling  process  has  its  breaking  strain  greatly  in- 
creased by  being  made  extremely  hard,  and  not  by  being  "  consolidated,"  as  pre- 
viously supposed. 

46.  Specimens  cut  out  of  crank-shafts  are  improved  by  additional  hammering. 

47.  The  galvanizing  or  tinning  of  iron  plates  produces  no  sensible  effects  on 
plates  of  the  thickness  experimented  on.     The  result,  however,  may  be  different, 
should  the  plates  be  extremely  thin. 

48.  The  breaking  strain  is  materially  affected  by  the  shape  of  the  specimen. 
Thus  the  amount  borne  was  much  less  when  the  diameter  was  uniform  for  some 


Art.  32.]  KIRKALDY'S  CONCLUSIONS.  2?$ 

inches  of  the  length  than  when  confined  to  a  small  portion — a  peculiarity  previously 
unascertained,  and  not  even  suspected. 

49.  It  is  necessary  to  know  correctly  the  exact  conditions  under  which  any 
tests  are  made  before  we  can  equitably  compare  results  obtained  from  different 
quarters. 

50.  The  startling  discrepancy  between  experiments  made  at  the  Royal  Arsenal, 
and  by  the  writer,  is  due  to  the  difference  in  the  shape  of  the  respective  specimens, 
and  not  to  the  difference  in  the  two  testing  machines. 

51.  In  screwed  bolts  the  breaking  strain  is  found  to  be  greater  when  old  dies 
are  used  in  their  formation  than  when  the  dies  are  new,  owing  to  the  iron  becoming 
harder  by  the  greater  pressure  required  in  forming  the  screw  thread  when  the  dies 
are  old  and  blunt  than  when  new  and  sharp. 

52.  The  strength  of  screw-bolts  is  found  to  be  in  proportion  to  their  relative 
areas,  there  being  only  a  slight  difference  in  favor  of  the  smaller  compared  with  the 
larger  sizes,  instead  of  the  very  material  difference  previously  imagined. 

53.  Screwed  bolts  are  not  necessarily  injured,  although  strained  nearly  to  their 
breaking  point. 

54.  A  great  variation  exists  in  the  strength  of  iron  bars  which  have  been  cut  and 
welded  ;  whilst  some  bear  almost  as  much  as  the  uncut  bar,  the  strength  of  others  is 
reduced  fully  a  third. 

55.  The  welding  of  steel  bars,  owing  to  their  being  so  easily  burned  by  slightly 
overheating,  is  a  difficult  and  uncertain  operation. 

56.  Iron  is  injured  by  being  brought  to  a  white  or  welding  heat,  if  not  at  the 
same  time  hammered  or  rolled. 

57.  The  breaking  strain  is  considerably  less  when  the  strain  is  applied  suddenly 
instead  of  gradually,  though  some  have  imagined  that  the  reverse  is  the  case. 

58.  The  contraction  of  area  is  also  less  when  the  strain  is  suddenly  applied. 

59.  The  breaking  strain  is  reduced  when  the  iron  is  frozen  ;  with  the  strain 
gradually  applied,  the  difference  between  a  frozen  and  unfrozen  bolt  is  lessened,  as 
the  iron  is  warmed  by  the  drawing  out  of  the  specimen. 

60.  The  amount  of  heat  developed  is  considerable  when  the  specimen  is  sud- 
denly stretched,  as  shown  in  the  formation  of  vapor  from  the  melting  of  the  layer 
of  ice  on  one  of  the  specimens,  and  also  by  the  surface  of  others  assuming  tints  of 
various  shades  of  blue  and  orange,  not  only  in  steel,  but  also,  although  in  a  less 
marked  degree,  in  iron. 

61.  The   specific  gravity  is  found  generally  to   indicate  pretty- correctly  the 
quality  of  specimens. 

62.  The  density  of  iron  is  decreased  by  the  process  of  wire-drawing,  and  by  the 
similar  process  of  cold  rolling,  instead  of  increased,  as  previously  imagined. 

63.  The  density  in  some  descriptions  of  iron  is  also  decreased  by  additional  hot- 
rolling  in  the  ordinary  way  ;  in  others  the  density  is  very  slightly  increased. 

64.  The  density  of  iron  is  decreased  by  being  drawn  out  under  a  tensile  strain, 
instead  of  increased,  as  believed  by  some. 


276 


CAST  IRON  IN   TENSION. 


[Art.  33. 


65.  The  most  highly  converted  steel  does  not,  as  some  may  suppose,  possess  the 
greatest  density. 

66.  In  cast  steel  the  density  is  much  greater  than  in  puddled  steel,  which  is 
even  less  than  in  some  of  the  superior  descriptions  of  wrought  iron. 


Art.  33.— Cast  Iron. 

Coefficient  of  Elasticity  and  Elastic  Limit. 

Cast  iron  is  a  material  of  much  less  value  to  the  engineer 
than  wrought  iron,  and  consequently  has  been  the  subject  of 
much  less  experimental  investigation. 

The  following  table  (Table  I.)  contains  values  of  the  co- 
efficient of  tensile  elasticity  for  three  (Nos.  I,  2  and  3)  differ- 
ent irons  used  in  the  fabrication  of  cast-iron  cannon.  They 
are  computed  by  the  aid  of  Eq.  (i),  Art.  2,  from  data  con- 
tained in  "  Reports  of  Experiments  on  the  Properties  of 
Metals  for  Cannon,"  etc.,  by  the  late  Captain  T.  J.  Rodman, 

TABLE   I. 


NO.  I. 

NO.  2. 

NO.  3. 

NO.  4. 

w. 

E. 

E. 

E. 

E. 

1,000 

28,011,000 

50,000,000 

33,333,000 

25,000.000 

2,000 

28,011,000 

28,571,000 

28,571,000 

16,667,000 

3,000 

25,000,000 

23,810,000 

27,273,000 

15,000,000 

4,000 

22,962,000 

22,727,000 

25,000,000 

15,385,000 

5,000 

23,031,000 

20,833,000 

23,810,000 

13,889,000 

10,000 

20,960,000 

17,000,000 

20,000,000 

12,195,000 

15,000 

16,773,000 

13,204,000 

17,241,000 

10,000,000 

20,000 

13,384,000 

7,370,000 

14,085,000 

8,000,000 

24,000 

10,150,000 

3,454,000 

II,O6O,OOO 

W  and  E  are  expressed  in  pounds  per  square  inch. 

U.  S.  A.  The  iron  was  an  excellent  charcoal  gun  iron,  and  the 
specimens  were  from  30  to  35  inches  long  turned  to  a  diameter 
of  1.382  inches.  The  data  were  selected  at  random  (pages  158, 


Art.  33.] 


ELASTICITY. 


277 


212  and  228  of  the  work  cited)  from  the  large  amount  accumu- 
lated by  Captain  Rodman. 

Column  No.  4  contains  values  of  E  given  by  Wm.  Kent, 
M.  E.  (Van  Nostrand's  Magazine,  Vol.  20) ;  they  belong  to  a 
piece  of  cast  iron  i-J  inches  in  diameter  and  5  inches  long. 

The  left-hand  column,  headed  "  W"  gives  the  stress  per 
square  inch,  while  the  three  columns  "£"  give  the  correspond- 
ing ratios  between  stress  and  strain  for  the  three  different 
irons.  Such  ratios  are  the  "  coefficients  of  elasticity,"  properly 
speaking,  below  the  elastic  limit  only.  It  will  be  observed, 
however,  that  none  of  these  specimens  can  really  be  con- 
sidered to  possess  an  elastic  limit,  unless  possibly  No.  I, 
wrhose  elastic  limit  may  be  taken  at,  or  a  very  little  above 
2,000  pounds  per  square  inch. 

In  No.  I  first  permanent  set  was  observed  at  4,000  pounds  per  square  inch. 
In  No.  2  first  permanent  set  was  observed  at  4,000  pounds  per  square  inch. 
In  No.  3  first  permanent  set  was  observed  at  8,000  pounds  per  square  inch. 


20000. 


15000. 


10000. 


5000 


1  inch 


2   nches 


3  ir.chc 


Fig.  I  represents  graphically  the  results  of  the  experiments 


2/8 


CAST  IKON  iff  TENSION: 


[Art.  33. 


on  specimen  No.  2.  The  constantly  varying  value  of  the  ratio 
between  stress  and  strain  is  shown  in  a  very  evident  manner 
by  the  continually  varying  inclination  of  the  curve.  The 
strains  (stretches)  are  laid  down  as  if  belonging  to  a  bar  1,000 
inches  long. 

The  following  results  are  deduced  by  B.  B.  Stoney  (Theory 
of  Strains  in  Girders  and  similar  Structures,  p.  369)  from  ex- 
periments by  Eaton  Hodgkinson  on  a  bar  of  English  cast  iron 
10  feet  long. 

W  =    2,240  pounds  per  square  inch E  =  13,603,520  pounds  per  square  inch. 


W=  4,480 
W—  6,720 
W  —  8  960 

(  ft  K  t 

«  11  fl  « 

E  —  13,260,800 
E  =  12,382,720 
..E  —  ii  ^06  480 

• 

! 

< 

W  =  11,200 

"W  —  i  a  4J.O 

(  «<  «  ( 

E  —  10,843,840 
E  —  9  856  ooo 

t    t 

( 

j 

W  —  14,560 

E  —  9,549,120 

These  results  show  a  limit  of  elasticity  at  about  6,000 
pounds  per  square  inch ;  they  also  show  much  smaller  values 
of  E  than  those  given  in  Table  I.  This  last  disagreement  is 
undoubtedly  due,  to  a  great  extent,  to  the  fact  that  the  values 
of  E  in  Table  I.  probably  all  belong  to  fine  charcoal  iron  fabri- 
cated for  a  special  purpose,  while  the  others  do  not. 

If  A  —  extension,  or  stretch  in  inches  of  a  cast-iron  bar 
when  acted  upon  by  a  force  W  (in  pounds),  and  if  /  represents 
the  length  of  the  bar  in  inches,  Mr.  Hodgkinson  deduced  the 
following  formulae  from  his  experiments : 


A  =  /{. 00239628  —  A/.OOOOO5742I5  —  .000000000343946^}  .  (i) 
For  bars  10  feet  long: 

Permanent  set,  in  inches  =  .0193^  +  .64^ (2) 

Although  the  preceding  results  are  only  a  few  of  a  great 


Art.  33.]  ULTIMATE  RESISTANCE.  279 

many  similar  results  that  may  be  computed  in  the  same  man- 
ner, yet  they  give  a  fair  representation  of  the  general  character 
of  the  elastic  properties  of  cast  iron.  The  metal  is  seen  to  be 
very  irregular  and  unreliable  in  its  elastic  behavior.  A  large 
portion  of  the  material  can  scarcely  be  said  to  have  an  elastic 
limit,  although  no  apparent  permanent  set  takes  place  under  a 
considerable  intensity  of  stress  ;  in  other  words,  although  per- 
haps all  tested  specimens  resume  their  original  shape  and 
'dimensions  for  small  intensities  of  stress,  yet  the  ratio  between 
stress  and  strain  is  seldom  constant  for  essentially  any  range 
of  stress. 

Ultimate  Resistance. 

On  page  5  of  Captain  Rodman's  "  Reports  "  are  given  the 
following  densities  and  ultimate  tensile  resistances,  expressed 
in  pounds  per  square  inch,  of  16  specimens  of  warm-blast, 
charcoal  Greenwood  and  Salisbury  iron,  taken  from  preliminary 
castings  of  second  and  third  fusion  pigs : 


DENSITY.                               ULT.    RESIST.  DENSITY.                                ULT.    RESIST. 

7-184 33,079  7-2io 22,547 

7-198 3i,3S4  7-I72 28,518 

7-307 35.4S6  7.159 36,573 

7-099 23,776  7.137 33,268 

7-304 3i,3J7  7-io6 22,290 

7-273 42,884  7.100 22,179 

7.272 38,993  7-iog 22,888 

7-219 25,372  7-191 23,873 


Again,  Table  II.  is  taken  from  page  261  of  the  same  "  Re- 
ports." The  results  are  for  specimens  from  trial  castings  of 
second-fusion  pigs.  The  ultimate  resistance  is  in  pounds  per 
square  inch,  while  the  strains  are  for  an  inch  of  length. 

"  Ult.  Ext"  is  the  ultimate  extension,  or  stretch,  just  be- 
fore fracture,  for  one  lineal  inch.  The  specimens  were  30 
inches  long  and  1.382  inches  in  diameter. 


280 


CAST  IRON  IN   TENSION. 


[Art.  33, 


TABLE   II. 


SPECIMEN. 

DENSITY. 

ULT.    EXT. 

ULT.    RESIST. 

Ao 

7.267 

.00303 

30,H7 

Ai 

7-274 

•00334 

31,681 

Bo 

7.178 

.00291 

23,617 

Bi 

7.202 

.COl6l 

24,260 

Co 

7-255 

.OO287 

28,22O 

Ci 

7.280 

.00382 

27,147 

Do 

7.221 

.00424 

25,627 

Di 

7.230 

.00223 

24,767 

On  page  42  of  "  Reports  of  Experiments  on  Metals  for 
Cannon,"  Major  Wade  gives  the  following  for  15  proof  bars 
cast  with  8-  and  lO-inch  guns  and  6-pounder  trial  guns,  at  South 
Boston,  1844 : 

Greatest  resistance  per  square  inch 31,027  pounds. 

Mean  "  "         "         "     27,232      " 

Least  "  "        "         " 22,402      " 

He  states  that  these  specimens  show  the  general  quality  of 
the  iron  used  at  that  time. 

Again,  on  page  179  of  the  same  "  Reports,"  Major  Wade 
gives  for  25  specimens  from  32-pounder  cannon  made  at  West 
Point  foundry  in  1850: 

Greatest  resistance  per  square  inch 36,728  pounds. 

Mean  "  "         "         "     32,023 

Least  "  "        "         "     28,990 

He  states  that  the  character  of  this  iron  was  "  that  of  good 
foundry  iron,  of  the  different  grades  of  Numbers  I,  2,  and  3  ; " 
it  was  composed  of  first,  or  first  and  second  fusion  pigs. 

The  preceding  results  give  correct  representations  of  the 
character  of  the  best  quality  of  American  cast  iron,  produced 
for  use  in  cases  requiring  such  a  metal. 


Art.  33.]  UL  TIM  A  TE  RE  SIS  TANCE.  2  8  1 

Three  specimens,  turned  down  to  a  diameter  of  about 
0.625  inch,  taken  from  the  iron  used  in  the  Boston  water 
mains,  and  broken  at  the  Warren  Foundry,  Phillipsburg,  N.  J., 
gave  the  following  ultimate  resistances  in  pounds  per  square 
inch  : 

15,470  ............     18,300 


As  with  all  material,  the  character  of  cast  iron  affects,  to  a 
great  extent,  its  resistance  ;  i.  ^.,  whether  it  is  fine  or  coarse 
grained,  gray  or  white,  etc.  It  (the  resistance)  also  depends 
upon  the  character  of  the  ore  from  which  it  is  produced. 

Major  Wade  ("  Reports,"  pages  378  and  388)  shows  that 
the  cold-blast  iron  which  he  tested  gave  much  higher  resist- 
ance than  the  hot-blast  metal. 

It  is  to  be  remembered  that  all  the  specimens  from  which 
the  preceding  results  were  deduced  were  what  may  be  called 
"  small  specimens."  Specimens  with  several  square  inches  in 
area  of  normal  section  would  probably  give  somewhat  different 
results. 

It  is  interesting  to  observe  that,  in  experimenting  upon 
cast-iron  cannon,  Major  Wade  ("  Reports,"  pages  77  and  78) 
found  that  water  was  forced  through  the  "  pores  "  of  the 
metal  of  one  cannon  at  a  pressure  of  7,000  pounds  per  square 
inch,  and  through  those  of  another  with  thicker  metal  (thick- 
ness equal  to  radius  of  bore)  at  a  pressure  of  9,000  pounds  per 
square  inch. 

Capt.  Rodman  ("  Reports,"  page  262)  forced  water  through 
the  pores  of  the  metal  of  cylinders  5  inches  long,  I  inch  thick, 
and  i  inch  bore,  at  pressures  ranging  from  15,276  to  25,464 
pounds  per  square  inch. 

The  experiments  of  Eaton  Hodgkinson  •  ("  Experimental 
Researches  on  the  Strength  and  Other  Properties  of  Cast 
Iron  "),  on  English  metal  gave  the  following  resistances  in 
pounds  per  square  inch  : 


282  CAST  IRON  IN   TENSION.  [Art.  33. 

Cannon  iron  No.  2,  hot  blast 13, 505  pounds. 

Cannon  iron  No.  2,  cold  blast 16,683  " 

Cannon  iron  No.  3,  hot  blast 17,755  " 

Cannon  iron  No.  3,  cold  blast 14,200  " 

Devon  (Scotland)  iron  No.  3,  hot  blast 21,907  " 

Buffery  iron  No.  I,  hot  blast 13,434  " 

Buffery  iron  No.  I,  cold  blast 17,466  " 

Coed-Talon  iron  No.  2,  hot  blast 16,676  " 

Coed-Talon  iron  No.  2,  cold  blast 18,855  " 

Low  Moor  iron  No.  3 14, 535  " 

Mixture 16,542  " 

Several  of  these  results  are  the  means  of  those  of  a  number 
of  tests.  The  areas  of  the  normal  sections  of  the  test  speci- 
mens varied  from  1.54  inches  to  4.27  inches,  being  considerably 
larger  than  those  of  the  specimens  tested  by  Major  Wade  and 
Captain  Rodman. 

The  characteristic  fracture  of  cast  iron  is  granular  and  crys- 
talline, with  very  little  (scarcely  perceptible  by  the  unaided 
eye)  reduction  of  area  or  elongation.  Fracture  takes  place 
suddenly  and  without  warning,  and  its  ultimate  resistance  is 
influenced  by  many  causes  whose  action  may  not  be  observed 
by  any  ordinary  means ;  for  these  reasons,  it  is  a  treacherous 
and  unreliable  material  in  tension,  as  indeed  any  brittle  ma- 
terial must  be. 


Effect  of-  Remelting. 

Crude  pigs  are  said  to  be  u  first-fusion  "  metal. 
Once  remelted  pigs  produce  "  second-fusion  "  iron. 
Twice  remelted  pigs  produce  "  third-fusion  "  iron. 

etc.,  etc.,  etc. 

On  page  237  of  Major  Wade's  '*  Reports,"  the  following 
values  are  given  for  Greenwood  first-fusion  iron  (iron  in  orig- 
inal pigs) : 


Art.  33.]  EFFECT  OF  REMELTJNG.  283 

ULT.  RESIST.  IN  POUNDS 
PER  SQ.  IN. 

No.  I  iron 15,129     (mean  of  3  tests). 

No.  2  iron 27,I53     (mean  of  2  tests). 

No.  3  iron 34,923     (mean  of  4  tests). 

"  No.  i  is  the  softest  gray  iron, 

"  No.  2  is  intermediate, 

"  No.  3  is  the  hardest  gray  iron." 

Again  on  page  240  : 

ULT.  RESIST. 

Greenwood,  No.  i,  1st  fusion 20,900  pounds  per  sq.  in. 

Greenwood,  No.  I,  2d  fusion 30,229  pounds  per  sq.  in. 

Greenwood,  No.  i,  3d  fusion 35,7^6  pounds  per  sq.  in. 

Guns  cast  from  3d  fusion 33, 815  pounds  per  sq.  in. 

The  last  result  is  a  mean  of  four  tests. 
Finally  on  page  242  : 

Nos.  i  and  2  mixed \  2d  fusion 2?'588  Pounds  Per  S*  |n' 

(  3d  fusion 40, 98 7  pounds  per  sq.  in. 

Nos.  i,  2,  and  3  mixed,  j  2d  fusion 37'789  P°unds  Per  S*  in" 

(  3d  fusion 32,485  pounds  per  sq.  in. 

It  is  seen  that  "  the  softest  kinds  of  iron  will  endure  a 
greater  number  of  meltings  with  advantage,  than  the  higher 
grades."  The  greatest  ultimate  resistance,  in  pounds  per 
square  inch,  is  obtained  with  : 

No.  i  iron  at  the  4th  fusion, 

Nos.  I  and  2  mixed  at  the       3d  fusion, 
Nos.  i,  2  and  3  mixed  at  the  2d  fusion. 

These  results  probably  indicate  about  the  limits  to  which 
the  remelting  of  this  iron  could  be  advantageously  carried. 

On  page  279  of  the  same  "  Reports/'  is  given  the  result  of 
the  test  of  a  specimen  of  third-fusion  iron,  of  a  mixture  of  Nos. 
i,  2  and  3,  taken  from  a  gun.  The  ultimate  resistance  found 


284 


CAST  IRON  IN   TENSION. 


[Art.  33. 


was  45,970  pounds  per  square  inch ;  a  most  remarkable  speci- 
men of  cast  iron. 

Effect  of  Continued  Fusion. 

Major  Wade  ("  Reports,"  pp.  38-41)  tested  the  effects  of 
continued  fusion  on  different  grades  of  iron,  both  in  relation  to 
transverse  and  tensile  resistance. 

The  general  result  was  an  increase  of  tensile  -resistance  up 
to  3^  hours  in  fusion,  which  was  the  longest  period  tried. 

The  following  results  are  taken  from  pp.  40  and  41  of  the 
"  Reports." 


TIME    IN    FUSION. 


ULT.    RESIST. 


Stockbridge , 


r  NO.  5. 

N 


Proof  bars. 


io-inch  Howitzer,   2d 
fusion  from  pigs. 


0.5  h( 

I.O 

1-5 

2.O 
0-5 
1-5 
3-0 

3-75 

0.00 
1.  00 
2.OO 
1.00 

inds  p 

er  squ 

are  in 

ch. 

.  .            2O  127 

24  387 

.  14  406 

25,969 

29,143 

27  7^^ 

1O  O1Q 

15  86r 

20,420 

24,383 
.  2=;.  771 

«    i<    < 

« 

These  tests  show  well  the  effect  of  continued  fusion  for  a 
period  not  exceeding  3.75  hours. 

Effect  of  Repetition  of  Stress. 

Capt.  Rodman  ("  Reports,"  p.  262)  experimented  on  the 
effect  of  repeated  stresses  with  the  following  results  : 


SPECIMEN. 


Ao  broke  at  23Oist  repetition  of  22,000    pounds  per  square  inch. 


Ao 
Bo 
Bo 
Co 
Co 
Do 


282d 


150th 
65ist 
457th 
I72d 


26,000 
20,000 
20,000 
22,500 
23,500 
21,600 


THE  f 

UNIVERSIT 


Art.  33.]  REPETITION  OF  STRESS. 

The  repetition  of  the  letters  representing  the  specimen 
indicates  that  duplicates  were  tested. 

A  reference  to  Table  II.  will  show  what  single  loads  per 
square  inch  broke  the  same  irons,  and  a  comparison  of  the  two 
will  exhibit  the  "  fatigue  "  of  the  metal. 

On  pages  166  and  167  he  also  gives  some  very  interesting 
results  of  intermittent  repetitions  of  stresses.  He  subjected  a 
cylinder  of  cast  iron,  1.382  mches  in  diameter  and  35  inches 
long  to  intermittent  repetitions  of  15,000  pounds  per  square 
inch  (about  three-quarters  of  its  ultimate  resistance)  as  follows: 
250  repetitions,  then  a  rest  of  40  hours  ;  next,  375  additional 
repetitions,  then  a  rest  of  30  days;  next,  155  additional  repe- 
titions, then  a  rest  of  29  days  ;  next,  1,020  additional  repe- 
titions, then  a  rest  of  26  days;  finally,  156  additional  repe- 
titions followed  by  breakage  at  the  1,  956th  repetition,  in 
every  case  "  rest  "  signifies  entire  freedom  from  load.  Capt. 
Rodman's  table  gives  a  detailed  account  of  these  .experiments. 
He  remarks  upon  them  as  follows  :  "  The  most  interesting 
point  ...  is  the  fact  that  at  every  interval  of  rest,  of  any 
considerable  time,  the  permanent  set,  and  the  extension  due  to 
the  last  previous  application  of  the  force,  diminished.  And  in 
some  instances  it  required  some  fifty  repetitions  to  bring  up 
the  extension  and  set  to  the  same  points  where  they  had  been 
at  the  beginning  of  the  period  of  rest  ;  thus  indicating  clearly 
that  the  specimen  was  partially  restored,  by  the  interval  of 
rest,  from  the  injury  which  it  had  received  ;  and  that  it  endured 
a  greater  number  of  repetitions,  owing  to  the  intervals  of  rest, 
than  it  would  have  done  had  the  repetitions  succeeded  each 
other  continuously,  and  at  short  intervals  of  time." 

These  experiments  show  the  '*  fatigue  "  of  cast  iron  and 
the  increase  of  the  ratio  of  stress  over  strain  produced  by 
"  rest  "  —  so  far  as  tensile  stress  is  concerned. 

An  examination  of  the  tables  also  shows  that  in  any  series 
of  repetitions,  between  any  two  consecutive  rests,  both  the 
extension  and  set  were  constantly  increasing,  consequently, 


286 


STEEL  IN   TENSION. 


[Art.  34. 


that  the   ratio  of   stress  over   strain   was    constantly  decreas- 
ing. 

Effect  of  High  Temperatures. 

A  few  experimental  results  bearing  on  this  point  will  be 
found  in  Table  IX.  of  Article  35. 


Art.  84.— Steel. 
Coefficient  of  Elasticity. 

The  great  number  of  the  varieties  and  grades  of  "  steel  " 
renders  possible  the  existence  of  a  correspondingly  great  num- 
ber of  the  mechanical  quantities  and  coefficients  used  in  its 
consideration  in  connection  with  the  "  Resistance  of  Mate- 
rials." In  every  case,  therefore,  the  kind  and  character  of  the 
steel  on  which  experiments  are  made,  should  be  stated.  In 
some  cases,  however,  this  is  impossible. 

TABLE  A. 


fc 
O 

COEFFICIENT  OF  ELASTICITY. 

MAKERS. 

z 

REMARKS. 

*. 

Greatest. 

Mean. 

Least. 

Coleman,  Rahm  &  Co.,  Pitts- 

burg1 

2 

4 

31,000,000 
20,325,000 

29,500,000 
25,600,000 
15,683,400 

2o,OOO,OOO 

13,324,600  \ 

"  Very  poor  steel." 
From    chrome   steel 

Am.  Tool  Steel  Co.,  Brooklyn 
Butcher  &  Co.,  Philadelphia 

;; 

4 
3 
•5 

24,200,000 
22,600,000 
32,150,000 

17,154,000 
17,298,000 
22,835,000 

11,950,000  J 
11,115,000 
17,605,000 

staves. 
Chrome  steel  from  ingot. 
Carbon  rivet  steel. 

6 

25,100,000 

21,145,000 

17,350,000 

Carbon  steel. 

i>                              it 

4 

25,700,000 
A 

22,712,000 
bout  27,000 

21,400,000 

000 

Carbon  steel. 
Carbon  steel  staves. 

Table  A  contains  coefficients  of  tensile  elasticity  for  the 
different  grades  of  steel  shown.  These  results  were  obtained 
from  tests  made  in  connection  with  the  construction  of  the 


Art.  34.] 


COEFFICIENT  OF  ELASTICITY. 


287 


St.  Louis  steel  arch,  and  have  been  taken  from  Prof.  Wood- 
ward's history  of  that  structure. 

These  coefficients  were  determined  for  cylindrical  speci- 
mens varying  from  0.5  inch  to  i.oo  inch  in  diameter  and  3.00 
to  6.00  inches  in  length. 

In  Table  I.  are  contained  the  coefficients  of  elasticity  of 
the  hardened  and  tempered  steel  wire  (see  Table  XXII.),  sup- 
plied by  the  different  makers  named,  in  response  to  the  call 
for  bids  for  the  steel  cable  wire  for  the  New  York  and  Brook- 
lyn suspension  bridge.  (Washington  A.  Roebling's  "  Report," 
1st  Jan.,  1877). 

In  the  same  "  Report,"  page  72,  the  specifications  state : 
"  The  elastic  limit  must  be  no  less  than  -ffa  of  the  breaking 
strength.  .  .  .  Within  this  limit  of  elasticity,  it  must 
stretch  at  a  uniform  rate  corresponding  to  a  modulus  of  elas- 
ticity of  not  less  than  27,000,000  nor  exceeding  29,000,000." 


TABLE   I. 


COEFFICIENTS 

OF  ELASTICITY. 

NUMBER  OF 

Greatest  E. 

Least  E. 

TESTS. 

J.  Lloyd  Haigh    

20,817,067 

28,815,797 

12 

Cleveland  Rollin^  Mills     

30  142  026 

28,QI7,7I1? 

6 

^Vashburne  &  Moen 

2Q  7^7  ^OO 

28,887  006 

6 

Sulzbacher,  Hymen,  "Wolff  &  Co  

30,380,046 

29,103,238 

6 

Tno.  A   Roebling's  Sons  Co  

30,231,020 

28,788,619 

13 

Carey  &  Moen                               .  . 

31  26l  O4I 

2O  4l8  O25 

12 

Table    I.    gives   the   greatest    and   least    results   of   these 
tests  in  pounds  per  square  inch,  in  the  columns  headed  4<  E" 


288 


STEEL  IN   TENSION. 


[Art.  34. 


together  with  the  number  of  tests  of  the  product  of  each 
maker.  -All  the  wire  was  No.  8,  Birmingham  gauge ;  i.  e., 
0.165  inch  in  diameter. 

It^not  evident  from  the  "  Report  "  whether  these  values 
were  obtained  for  some  particular  intensity  of  stress,  or 
whether  they  are  mean  values  for  the  entire  range  below  the 
elastic  limit. 

TABLE   II. 


KIND   OF  STEEL. 

SPEC. 
GRAV. 

CARBON. 

SET. 

E. 

SET. 

E'. 

Hammered  Bessemer  from 
Ho<rbo  (round) 

7  8^2 

I     ^ 

o  003 

<?y    8^Q   68O 

Hammered  Bessemer  from 
Hoo'bo  (square)  

7.8<;o 

1.26 

O.OO4 

OQ  124  1  80 

o  006 

1O  Hm  QOO 

Hammered  Bessemer  from 
Hogbo  (square)  

7.840 

I.O5 

0.014 

•30,604,^20 

o.ooo 

•3,1  406  «;8o 

Rolled  cast  steel  from  Wik- 
manshyttam  (round)   .  .  . 
Hammered  cast  steel  from 
F    Krupp  (round).  .      .  . 

7.832 
7.841 

1.22 

0.61 

0.021 

O.OOO8 

31,222,100 
a  I  •}CQ,'}4O 

o  004 

32  114  160 

Rolled  puddled  steel  from 
Surahammar  (square).  .  . 
Rolled  puddled  steel  from 
Surahammar  (square).  .  . 

7.781 
7.828 

0.66 
0.56 

0.027 

29,918,320 

0.015 

30,330,040 

Table  II.  contains  coefficients  of  tensile  elasticity  for  the 
steels  named,  as  determined  by  Knut  Styffe  ("  Iron  and  Steel," 
pages  146  and  147).  He  pursued  the  following  method  :  Let 
/'  and  /  represent  the  stretch,  or  strain,  for  unit  of  length  of  a 
bar  for  the  two  intensities  /'  and  /.  By  the  principles  estab- 
lished in  Article  2  : 


=  £,    and    /'  =  £ 


Hence, 


Art.  34.]  COEFFICIENT  OF  ELASTICITY.  289 


Eq.  (i)  really  gives  a  kind  of  "  mean  "  coefficient,  for  it  is 
based  on  the  assumption  that  £  is  the  same  for  different  in- 
tensities of  stress.  It  will  be  seen  in  Table  III.  that  this  is 
sometimes  far  from  true. 

The  columns  "  carbon  "  and  "  set  "  contain  per  cents  of 
those  quantities.  E  is  the  coefficient  of  elasticity  before  the 
bar  is  heated,  and  E'  the  same  quantity  after  the  bar  had  been 
heated  to  "  slight  redness."  "  Set  "  is  the  permanent  elonga- 
tion (in  per  cents  of  original  lengths)  just  before  E  or  E'  was 
measured.  The  test  specimens  were  small  bars  varying  in 
original  area  from  0.1015  square  inch  to  0.2065  square  inch. 

The  experiments  of  Styffe  showed  that  "  by  such  mechani- 
cal operations  as  stretching,  hammering,  etc.,"  the  coefficient 
of  elasticity  may  be  diminished  ;  "  whilst  by  a  moderate  heat, 
or  still  better  by  a  glowing  heat,  it  may  be  increased." 

Table  III.  has  been  computed  from  data  obtained  by 
David  Kirkaldy  during  his  experiments  on  Fagersta  steel 
plates  ("  Experimental  Inquiry  into  the  Properties  of  Fagersta 
Steel,"  series  D,  Part  i).  The  test  specimens  were,  in  the 
clear,  2]^_  inches  wide  and  100  inches  long.  The  thickness  is 
given  in  the  horizontal  row,  as  shown.  The  values  of  the  co- 
efficient of  elasticity  (E)  are  the  greatest  and  least,  in  pounds 
per  square  inch,  for  the  various  intensities  "/,"  for  five  unan- 
nealed  J*j,  J^,  ^,  y2  and  ^-inch  (nominally)  plates  and  five 
similar  annealed  ones. 

These  show  very  irregular  elastic  behavior.  The  $fa  inch 
annealed  specimen  is  the  only  one  which  can  properly  be  con- 
sidered as  possessing  a  true  "  coefficient  of  elasticity  "  (about 
29,000,000  pounds  per  square  inch)  above  the  stress  intensity 
of  10,000  pounds,  the  ratios  of  stress  to  strain  are  so  very 
variable.  Prof.  Bauschinger's  "stretch-limit"  is  clearly  shown, 
for  the  different  specimens,  at  that  point  of  stress  where  the 
i9 


2QO 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE   III. 


UNANNEALED. 

ANNEALED. 

Greatest  E. 

Least  E. 

Greatest  E. 

Least  E. 

10,000 

45,455,ooo 

33,333,ooo 

37,037,000 

29,412,000 

14,000 

38,889,000 

30,435,000 

34,146,000 

29,167,000 

i8,ooo 

36,000,000 

29,032,000 

32,727,000 

29,032,000 

22,000 

34,375,ooo 

28,205,000 

31,429,000 

28,947,000 

26,000 

33,333,ooo 

25,000,000 

30,952,000 

20,968,000 

30,000 

31,915,000 

1,714,000 

29,412,000 

1,765,000 

34,000 

30,631,000 

1,107,500 

4,151,000 

1,066,000 

38,000 

29,008,000 

821,000 

1,214,000 

805,000 

42,000 

13,084,000 







46,000 

4,670,000 







Thickness. 

o.  125  inch. 

0.380  inch. 

0.255  inch. 

0.625  inch. 

values  of  E  are  almost  annihilated.  In  these  four  specimens 
the  first  permanent  sets  were  noted  at  40,000,  20,000,  30,000 
and  20,000  pounds  per  square  inch  respectively. 

In  1868  and  1870  the  "Steel  Committee"  of  the  British 
Institution  of  Civil  Engineers  made  some  valuable  experiments 
on  different  grades  of  steel.  The  following  values  of  E,  in 
pounds  per  square  inch,  are  computed  from  data  established 
by  that  committee  : 

Eighteen  specimens,  50  inches  long  and  1.382  inches  in  diame- 
ter, of  Bessemer  steel  tires,  axles  and  rails  (14  hammered  and  4 
rolled},  gave  : 


Greatest E  =  19,310,000  pounds  per  square  inch  \ 

Mean E  =  18,211,000  pounds  per  square  inch  V 

Least E  =  17,231,000  pounds  per  square  inch ) 


(2) 


Ten  hammered  samples  of  crucible  steel  tires,  axles  and  rails, 
and  one  of  rolled  axle,  all  with  preceding  dimensions,  gave  : 


Art.  34.]  COEFFICIENT  OF  ELASTICITY.  2QI 

Greatest E  =  19,310,000  pounds  per  square  inch  \ 

Mean E  =  17,778,000  pounds  per  square  inch  >•      .     .     (3) 

Least E  —  16,232,000  pounds  per  square  inch) 

All  these  are  evidently  for  very  soft  steels,  none  of  whose 
tensile  resistances  exceeded  91,700  pounds  per  square  inch. 

The  following  results  are  computed  from  samples  ten  feet 
long  and  one  and  one  half  inches  in  diameter: 

Eleven  samples  of  Bessemer  steel  gave  : 

Greatest E  —  29,867,000  pounds  per  square  inch  \ 

Mean E  =  28,718,000  pounds  per  square  inch  >      .     .     (4) 

Least E  —  27,317,000  pounds  per  square  inch ) 

Nineteen  specimens  of  crucible  steel  (chisel-rods,  gun-barrels, 
etc.),  gave  : 

Greatest E  =  29,867,000  pounds  per  square  inch  j 

Mean E  •=  28,718,000  pounds  per  square  inch  >•      .     .     (5) 

Least E  =  26,654,000  pounds  per  square  inch  ) 

These  last  were  much  harder  steels,  with  ultimate  resist- 
ance varying  from  75,300  to  118,300  pounds  per  square  inch. 

All  the  results  in  Eqs.  (2),  (3),  (4)  and  (5)  are  calculated  for 
the  strains  at  the  so-called  "  elastic  limit."  It  is  probable  that 
considerably  larger  values  would  be  obtained  for  the  ratio  (£) 
between  stress  and  strain  at  much  lower  intensities  of  stress. 

Prof.  Alex.  B.  W.  Kennedy  (London  "  Engineering,"  Vol. 
XXXI.,  1881)  determined  the  coefficients  of  tensile  elasticity 
of  specimens  of  mild  steel  plates  containing  about  0.18  per 
cent,  of  carbon,  and  of  some  specimens  of  still  milder  rivet 
steel. 

Twelve  specimens  of  plates  (i^xj^;  4xj^;2x^; 
3^4  x  y§;  and  2^  X  y2,  all  in  inches)  gave : 

GREATEST.                                                                     MEAN.  LEAST. 

33,670,000 29,882,000 25,440,000 

all  in  pounds  per  square  inch. 


STEEL  IN   TENSION.  [Art.  34. 

Eight  other  specimens  of  the  same  plates  gave  : 

GREATEST.                                                                    MEAN.  LEAST. 

3I,94O,OOO 29,OOI,OOO 26,IIO,OOO 

all  in  pounds  per  square  inch. 

As  a  rule,  the  thinner  plates  gave  the  higher  values  of  E. 
There  were,  however,  some  marked  exceptions. 

Eleven  specimens  of  \\  inch  round  rivet  steel,  turned  to  about 
-f  inch  diameter ;  two  each  of\\  and  ifa  inch  round,  turned  to  -J 
and  -J  inch  diameter,  respectively,  gave  : 

GREATEST.                                                                MEAN.  LEAST. 

3I,75O,OOO 3O,67O,OOO 


all  in  pounds  per  square  inch. 

Hay  Steel. 

Some  experiments  on  three  different  bars  of  the  Hay  steel 
used  in  the  bridge  at  Glasgow,  Missouri,  by  Gen.  Wm.  Sooy 
Smith,  gave  the  following  results  ("Annales  des  Fonts  et 
Chaussees/'  Feb.,  1881): 

Experiment  No.  I. 

A  bar  of  rectangular  section  2.09  x  i.i  inches,  reduced  by 
hammering  from  a  bar  2.6  inches  square,  was  subjected  to  dif- 
ferent intensities  of  stress  varying  from  about  20,500  to  54,000 
pounds  per  square  inch,  at  which  the  following  values  of  the 
coefficients  of  elasticity  (in  pounds  per  square  inch)  were 
found  : 

GREATEST.                                                                 MEAN.  LEAST. 

32,900,000 28,824,000 26,094,000 

At  54,000  pounds  per  square  inch  there  was  a  "  trace  "  only 


Art.  34.]  COEFFICIENT  OF  ELASTICITY.  293 

of  permanent  elongation  or  set.     The  length  of  this  bar,  be- 
tween the  observation  marks,  was  about  38.5  inches. 


Experiment  No.  2. 

A  round  bar  1.04  inches  in  diameter  was  subjected  to  a 
stress  of  about  51,200  pounds  per  square  inch,  with  a  stretch  of 
1.66  millimetres  per  metre,  at  which  a  "  trace  "  only  of  perma- 
nent set  was  observed.  The  resulting  coefficient  of  elasticity 
was: 

E  =  30,857,000  pounds  per  square  inch. 

The  distance  between  observation  marks  was  about  18.7 
inches. 

Experiment  No.  3. 

A  bar  5.2  x  1.34  inches  was  subjected  to  a  stress  of  about 
49,200  pounds  per  square  inch,  with  a  trace  only  of  permanent 
set  and  a  strain  of  0.00171  metre  per  metre.  Consequently 
the  resulting  coefficient  of  elasticity  was : 

E  =  28,764,000  pounds  per  square  inch. 

These  experiments  show  that  the  coefficient  of  elasticity  of 
Hay  steel  is  not  essentially  different  from  that  of  other  mate- 
rial of  the  same  class. 

In  his  "  Report  on  the  Renewal  of  Niagara  Suspension 
Bridge,"  Mr.  Leffert  L.  Buck,  C.  E.,  gives  the  following  values 
for  the  Hay  steel  used  in  that  work  : 

GREATEST.                                                                    MEAN.  LEAST. 

30,830,000 28,000,000 26,400,000 

all  in  pounds  per  square  inch.     These  results  are  for  eighteen 
experiments  on  small  specimens. 


294 


STEEL  IN   TENSION. 


[Art.  34. 


Ultimate  Resistance  and  Elastic  Limit. 

In  this  section,  it  is  to  be  observed,  the  "  elastic  limit  "  is 
seldom  that  point  at  which  the  coefficient  of  elasticity  (stress 
over  strain)  ceases  to  be  essentially  constant,  but  more  nearly 
Prof.  Bauschinger's  "  stretch-limit,"  at  which  the  increment  of 
strain,  due  to  a  constant  increment  of  stress,  very  suddenly 
increases,  involving  a  correspondingly  great  permanent  set. 

TABLE    Ilia. 


ULTIMATE  RESISTANCE,  POUNDS 

i 

PER  SQUARE   INCH. 

MAKERS. 

% 

REMARKS. 

H 

5 

Greatest. 

Mean. 

Least. 

Coleman,  Rahm  &  Co.,  Pitts- 

burg 

3 

i 
7 

2 

118,400 

144,300 
118,000 

91,200 
106,500 
112,100 
118,000 

74,000 
118,000 

"  Very  poor  steel." 
For  lathe  tools. 

Am.  Tool  Steel  Co.,  Brooklyn. 
Butcher  &  Co.,  Philadelphia.  . 
Park  Bros.,  Pittsburg  

Steel  Works,  New  York  

I 



85,400 



Jessup,  Sheffield,  Eng    

4 

86,000 

78,500 

74^00° 

j  2  Annealed. 
j  2  Unannealed. 

Anderson  &  Woods,  Pittsburg. 
Coleman,  Rahm  &  Co.,  Pitts- 

2 

100,000 

100,000 

100,000 

burg  

I 



68,000 



Miller,  Barr  &  Parkin,   Pitts- 

burg   

I 



90,000 

^_^_ 

Annealed. 

Miller,  Barr  &  Parkin,    Pitts- 

burg   

2 

103,200 

102,200 

101,200 

Unannealed. 

Huss^y,  Wells    &  Co.,    Pitts- 

burg   

3 
7 
3 
4 
4 

128,000 
81,300 
113,900 
110,100 
107,500 

126,500 
75,450 
112,600 
103,500 
106,000 

125,000 
45,000 

112,000 

99,200 
103,500 

Steel  plate. 
Cast  "machinery  steel." 
"  Gun  metal." 
Chrome  steel  stave. 
Chrome  steel  stave. 

Brown  &  Co.,  Pittsburg.  .  . 
Thos.  Firth,  Sheffield,  Eng.   . 
Butcher  &  Co.,  Philadelphia  . 

u 
u 

3 

I 

151,000 
129,000 
128,300 

148,700 
99,900 
98,300 

147,000 
69,800 
65,300 

Chrome  steel  ingot. 
Carbon  rivet  steel. 
Carbon  steel. 

u 

5 

142,000 

120,100 

100,000 

Carbon  rivet  steel. 

" 

5 

143,600 

136,300 

Chrome  steel  stave. 

n            n 

4 

135,400 

119,200 

111,700 

Chrome  steel  stave. 

N.  Y.  Chrome  Steel  Co  ". 
Parks  Bros.,  Pittsburgh  ^j- 

16 
3 

193,260 
131,864 

146,400 
119,500 

115,800 
109,500 

Chrome  steel  bar. 
Steel  normal  untemp. 

u           u                 u             tlj 

3 
3 
3 

227,500 
176,100 
150.500 

217,400 
165,500 
141,900 

201,341 

152,500 
132,700 

Temp,  in  oil  at  82"  F. 
Temp,  in  water  at  79°  F. 
Temp,  in  water  at  79°  F. 

Art.  34.] 


RAIL   STEEL. 


295 


Table  Ilia  is  condensed  from  Prof.  Woodbury's  history  of 
the  St.  Louis  arch.  The  last  four  results  are  from  the  experi- 
ments of  Chief  Engineer  Shock,  U.S.N.,  while  the  "  N.  Y. 
Chrome  Steel  Co."  result  is  from  Kirkaldy's  tests. 

The  diameters  of  the  (circular)  specimens  varied  from  0.357 
inch  to  i.oo  inch,  and  their  lengths  from  3  to  12  inches.  The 
elastic  limit  varied  from  45  to  55  (nearly)  per  cent,  of  the  ulti- 
mate resistance. 

TABLE   IV. 
Rail  Steel. 


NO. 

UNANNEALED. 

ANNEALED. 

CARBON, 
PERCENTS. 

T. 

E.  L. 

STR. 

T. 

E.L. 

STR. 

8 

79,625 

37,625 

IQ.6 

78,250 

35,750 

20.5 

0.324 

8 

8l,25O 

36,625 

15-6 

77,375 

37,125 

14.7 

0-379 

4 

72,750 

32,250 

22.5 

70,750 

29,250 

22.0 

0.308 

4 

76,750 

38,250 

n-5 

74,000 

30,500 

9-5 

0.438 

4 

75,750 

36,500 

12.8 

73,750 

34,000 

15.2 

0.405 

4 

83,500 

37,ooo 

14-5 

80,750 

40,000 

15-2 

0.384 

8 

72,375;- 

36,625 

17-5 

70,500 

33,875 

19.2 

0.282 

8 

79,875 

38,000 

14-5 

78,000 

36,875 

14.4 

0.381 

4 

70,500 

34,250 

17.0 

69,  500 

30,500 

17-5 

0.367 

4 

82,000 

40,250 

12.7 

76,000 

38,500 

13.2 

0-394 

4 

78,000 

36,750 

10.7 

74,250 

33,250 

13-7 

0.378 

4 

77,000 

40,250 

15-0 

76,000 

35,750 

14-5 

0.388 

24 

74,542 

35,833 

18.9 

72,958 

33,i67 

19.8 

0.314 

24 

80,167 

37,958 

14.1 

76,792 

36,167 

13-5 

0.392 

8 

76,875 

36,625 

ii.  7 

74,000 

33,625 

14.4 

0.391 

8 

80,250 

38,625 

14.7 

78,375 

37,875 

14.8 

0.3S6 

32 

75,^25 

36,031 

17-1 

73,219 

33,28i 

18.5 

0-334 

32 

80,188 

38,125 

14.2 

77,i88 

36,594 

13-8 

0.390 

Table  IV.  contains  the  results  of  one  hundred  and  ninety- 
two  tests  of  specimens  from  steel  rails  which  had  been  in  use 
on  the  Penn.  R.  R.  during  periods  of  time  of  different  lengths, 
These  results  were  given  by  Charles  B.  Dudley,  Ph.D.,  Chemist 
to  Penn.  R.  R.  Co.,  in  his  "  Report  to  the  Supt.  of  Motive 


296 


STEEL  IN   TENSION. 


[Art.  34. 


Power,"  and  published  in  the  Journal  of  the  Franklin  Institute, 
March,  1881. 

The  specimens  were  circular  and  turned  to  a  diameter  of 
0.75  inch  between  shoulders  five  inches  apart. 

The  following  is  the  notation  : 

"No"  is  the  number  of  specimens   for  which  the  other 

quantities  are  the  average. 
"  T"  is  the  ultimate  tensile  resistance  in  pounds  per 

square  inch. 

"  E.  L."  is  the  elastic  limit  in  pounds  per  square  inch. 
"  Str"  is  the  per  cent,   of  original  length  of  ultimate 

elongation  or  stretch  (/.  <?.,  at  instant  of  rupture). 


TABLE  V. 
Ba  rs   Una  nnea  led. 


CARBON. 

STRESS   IN   POUNDS   PER  SQUARE   INCH. 

PER  CENTS. 

Per  cent. 

Ultimate. 

Elastic  Limit. 

Str. 

Cont. 

0.30 
0.30 

94,760  1     £ 
95,380    1     uS 

55,712  1     ^ 
56,009         ^ 

'5-']    4 
12.9  I   3 

30.  1 
26.     1 

0 
CO 

0.30 

95,830    }•    <* 

55,120   ^  ^ 

15.3  1-  || 

II 

0.30 

96,020 

55,830        '1 

X4-5-            .; 

27-     I 

0.30 

94-970  j    ^ 

55,512  j    j> 

13-8    J      < 

29.   J 

' 

o 

0.50 

112,340   ]      0 

65-790!    I 

10.8  ^    1 

I9.   ] 

O 

0.50 

112,470     I       N 

66,O4O     |       x^ 

8.9      2 

16. 

M' 

o.  50 

III,980     J-     w 

66,160  }-  ^ 

10.5   Y    II 

22.     > 

II 

0.50 

113,320              II 

65,550  1 

10.9   1     . 

21. 

0.50 

113,040   j       5; 

65,980  J    >•' 

9-4  J    < 

20.    J 

4 

^ 

"  Str"  is  the  elongation  of  original  length. 
"  Cont"  is  the  contraction  of  original  section. 


Art. 


E  YE-BARS. 


297 


w    ^ 

J       >j 


S 


if, 

iff 

R       H       M 

III 

1  -  - 
«l  1 

o     -S     .S 

d    s    d 

I   f  I 

.s    .s    .s 

H      <      < 

!        H        M 

CO          CO         CO 

-•*          r*f 

•S         M         M 

•*••         •-*?>        raa> 

vn       vn       in 

*    3    H 

"o      "S      "o 

111 

III 

° 

M             <*           H 

K         ir>        cx 

OOO 

s  -  §  , 

§    £    o    « 

•tt  =  'AY 

S-K  =  -AY 

c< 

/•  A  N 

/  A  ^ 

1  s  °  2 

. 

o 

S  s  E  s 

S         ^          ^« 

ON        0 

O        a        rt- 

u 

1  °  ,(  „• 

i  -g  =  -AY 

Z'6  —  -AY 

£'Z  =  'AY 

£    £    2    2 

/  *  N 

*  *  N 

/  *  v 

<N        a        o 

C^            CO         CO 

O        0         « 

h      u 

«     <^    r; 

CO             O*           M 
M 

M      ci-     a 

« 

•oot-'o6  =  -AY 

•£oo'C6  =  -AY 

•031^9=  -AY 

S             s 

_^  

^_ 

_A  

I             1 

D    t;         ^ 

80      o 
0         0 
T}-        in        co 

t->i          !-«.         CO 

sO         m        i^>» 

ooo 

•<*•       Q        O 

M             O              Tf 

o      <             P 

•  0, 

C4              M              Tf 

X 

co         O^      co 

vO        *O        vO 

2     2 

!/)        H 

.ti 

•e8o'«  =  -AV 

•I8i^S  =  -AV 

^I9'9S  =  'AY 

W               Q, 

«    w         v3 

/  *  \ 

/•  *  \ 

/  A  ^ 

§              -B 
g             J 

H 

O          CO         CO 
tn       m       m 

N        xn      co 

O         O          M 

H*        rf        ci 
in       in       m 

co        0s       O 
r>.       m       M 

co"       ^T        in 
vn       in       m 

•aanxovj 
-nNvw  ao  aaow 

,,sdn 

•P9IIOH 

-WPAV 

o         '3l°H  «!d 

•^unnpu^HouHe 

z 

•9X3  SSOJOB  saqDut  ^^  '^pjm  s31!311!  fi 

•u,3JS 

•^aaj  01   x  ipui  f  x  saqoui  £ 

y 

i  1 

ooo               ooo               ooo 

cococo                    rococo                    rococo 

a  B 

666              666              666 

1 

298  STEEL  IN   TENSION.  [Art.  34. 

It  will  be  observed  that  the  process  of  annealing  decreased 
both  the  ultimate  resistance  and  elastic  limit.  The  results  are 
irregular,  however,  so  far  as  the  strains  or  elongations  are  con- 
cerned. 

Tables  V.  and  VI.  are  taken  from  "  Steel  in  Construction," 
a  paper  read  before  the  Engineers'  Society  of  Western  Penn- 
sylvania, by  Albert  F.  Hill,  C.  E.,  2Oth  April,  1880.  Table  V. 
contains  the  results  of  tests  of  specimen  bars  3"  X  7/z"  X  30". 
These  specimens  were  cut  from  rolled  bars  which  were  subse- 
quently manufactured  into  eye-bars. 

Nine  eye-bars  containing  0.30  per  cent,  of  carbon  were 
made,  besides  nine  others  containing  0.50  per  cent,  of  carbon. 
Each  group  of  nine  was  divided  into  three  classes,  with  welded, 
rolled  and  upset  heads,  respectively.  These  eighteen  eye-bars 
were  then  put  into  the  testing  machine,  and  the  results  belong- 
ing to  the  nine  containing  0.30  per  cent,  carbon  are  given  in 
Table  VI. 

The  results  of  the  tests  of  the  bars  containing  0.50  per 
cent,  carbon,  were  not  given,  but  it  was  stated  by  Mr.  Hill 
that  they  verified  the  conclusions  drawn  from  Table  VI. 

The  value  of  Table  VI.  is  much  enhanced  by  the  fact  that 
the  results  which  it  contains  belong  to  full-sized  bars,  and  not 
to  prepared  specimens.  The  bearing  of  the  information  which 
it  gives  on  the  mode  of  manufacture  of  the  eye-bar  head  is 
also  most  important.  It  will  be  observed  that  the  welded  head 
possesses  much  less  resistance  than  the  others,  and  that  the 
rolled  head  is  a  little  stronger  than  the  upset. 

Mr.  Hill  considered  that  these  experiments  "  clearly  estab- 
lish the  fact  that  welding  of  high-grade  steel  for  purposes  of 
construction  is  out  of  the  question  in  general  practice." 

The  metal  in  these  bars  was  "  open-hearth  "  steel  made  by 
Messrs.  Anderson  &  Co.,  of  Pittsburg,  Penn. 

Table  VI  I.,  gives  the  extension  of  the  results  on  bars  of 
Hay  steel,  a  part  of  which  have  already  been  given  in  the  sec- 
tion on  the  coefficient  of  elasticity.  The  "  elastic  limit "  is  the 


Art.  34.] 


HA  Y  STEEL  BARS. 


299 


intensity  of  stress  at  which  "  traces  "  of  permanent  set  begin 
to  be  observed,  or  immediately  before. 

TABLE   VII. 
Bars  of  Hay  Steel. 


POUNDS   OF   STRESS    PER 

PER   CENT.   OF 

LENGTH   IN 

SIZE   OF  BAR  IN 

SQ.   IN.   AT 

NO. 

INCHES. 

INCHES. 

Elastic 
Limit. 

Ultimate 
Resistance. 

Final  Stretch. 

Contraction. 

I 

38.5 

2.09  x  i.i 

54,ooo 

93,200 

10.00 



2 

I8.7 

1.04  o 

51,200 

98,100 

10.00 



3 



5.20  x   1.34 

49,200 

93,900 

8.00 

11.00 

4 



3.40  x   1.05 

54,400 

95,300 

12.00 

7.00 

5 



0.08  o 

— 

108,100 

6.00 

43-oo 

6 



1.04  o 

— 

128,700 

— 

35.00 

Bar  No.  I  was  hammered  down  from  a  square  bar  2.6  x  2.6 
inches  ;  and  No.  4  likewise  from  a  6.24  x  1.56  inch  bar. 

The  extension,  or  stretch,  in  No.  3,  was  taken  just  before 
failure. 

Experiment  No.  6  was  made  at  a  temperature  of  22°(Cent.?) 
below  zero. 

The  "  Report  on  the  Renewal  of  the  Niagara  Suspension 
Bridge/'  by  Mr.  Leffert  L.  Buck,  C.  E.,  1880,  has  already  been 
alluded  to,  in  connection  with  the  coefficient  of  elasticity. 
That  "  Report  "  contains  the  results  of  experiments  made  on 
plate  specimens,  of  the  Hay  steel  used  in  that  work,  with  cross- 
sectional  areas  varying  from  about  0.32  square  inch  to  0.72 
square  inch,  and  with  lengths  varying  from  seven  to  ten  inches 
between  clamps. 


300  STEEL  IN   TENSION.  [Art.  34. 

These  specimens  were  subjected  to  various  kinds  of  treat- 
ment, such  as  punching,  annealing,  blows  while  under  stress, 
nicking  on  edges,  etc.,  etc. 

One  specimen  was  also  subjected  to  intermittent  stresses. 

These  manipulations  necessarily  affected  the  elastic  limit 
and  ductility  as  well  as  the  ultimate  resistance.  With  the 
omission  of  the  one  specimen  just  mentioned,  the  extreme  re- 
sults are  as  follows  : 


Greatest.  .  . 

-Q       A  <3I 

Elastic  limit  • 

Least 

AT  aoo 

Greatest     . 

.    Q7  6oO 

Ultimate  resistance. 

Least      . 

CQ  "370 

Greatest  .  . 

.    IQ-4 

Final  stretch 

Least      .  .  . 

7.O 

Final  contraction  of 

.    42.O 

runtured  section.  . 

Least  .  . 

.    I^.O 

per  cent. 


The  "stretch"  and  "  contraction "  are  per  cents  of  ten 
inches  and  original  section,  respectively. 

Table  VIII.  contains  the  results  of  the  experiments  of  Sir 
Wm.  Fairbairn  on  the  different  varieties  of  English  steel  given 
in  the  left-hand  column.  The  specimens  were  one  inch  square, 
and  had  previously  been  subjected  to  a  transverse  load.  The  per 
cents  of  strain  or  elongation  are  for  a  length  of  eight  inches, 
which,  it  is  presumed,  included  the  section  of  fracture. 

Table  IX.  contains  the  results  of  tensile  experiments  on 
Bessemer  and  crucible  steel  specimen  bars  by  Mr.  Kirkaldy  for 
the  "  Steel  Committee  "  (English). 

The  first  part  of  the  table  gives  the  results  of  experiments 
on  bars  turned  accurately  to  a  diameter  of  1.382  inches  with  a 
length  "  in  the  clear  "  of  50  inches.  It  is  presumed  that  the 
per  cents  of  elongation  apply  to  that  length. 

The  second  part  of  the  table  gives  the  results  of  experi- 
ments on  bars  "  in  their  natural  skins,"  with  a  diameter  of  1.5 
inches  and  length  of  120  inches ;  to  which  length  the  per  cents 
of  elongation  apply. 


Art.  34.] 


ENGLISH  BAR   SPECIMENS. 


301 


TABLE   VIII. 
Bar  Specimens. — 1867. 


PRODUCERS. 


ULTIMATE   RESISTANCE 
PERSQ.   INCH. 


FINAL    STRAIN    OR 
ELONGATION. 


Messrs.  Brown  &  Co. 


Best  cast  steel,  for  turning  tools 

Best  cast  steel,  milder 

Cast  steel  from  Swedish  iron  for  tools  . 

Cast  steel,  milder,  for  chisels 

Cast  steel,  mild,  for  welding 

Bessemer  steel 

Double  shear  steel  from  Swedish  iron. . 

Foreign  bar,  tilted  direct 

English  tilted  steel 

C.  Cammel  &>  Co. 

Cast  steel,  termed  "  Diamond  Steel  ". . 

Cast  steel,  termed  "  Tool  Steel  " 

Cast  steel,  termed  "  Chisel  Steel" . 

Cast  steel,  termed  "  Double  Shear  Steel 

Hard  Bessemer  steel 

Soft  Bessemer  steel 

Messrs.  Naylor,   Vickers  6°  Co. 

Cast  steel,  called  "  Axle  Steel  " 

Cast  steel,  called  "  Tire  Steel  " 

"  Vickers  Cast  Steel,  Special  " 

"  Naylor  &  Vickers'  Cast  Steel  " 

S.  Osborne. 

Best  tool  cast  steel 

Best  chisel  cast  steel 

Sates-cup,  shear  blades,  etc 

Best  cast  steel  for  taps  and  dies 

Toughened  cast  steel  for  shafts,  etc.. . 

Best  double  shear  steel 

Extra  best  tool  cast  steel 

Boiler  plate,  cast  steel , 


Pounds. 
68,404 

106^714 

116,183 

110,055 

91,972 

92,555 

76,474 

59,538 


110,055 

109,072 

120,398 

96,665 

89,121 

81,483 


88,665 
91,520 

134,145 
118,066 


98,942 
123,686 
115,849 

98,790 
103,116 

87,931 

85,724 
111,676 


Per  cent. 

0.25 

1-5 
1.0 
3-62 

3.31 
19.62 

5-43 
13-56 
21.06 


1-53 
1.50 
2.50 

2-37 
20.87 

20.43 


16.25 
9.00 

1. 00 

1.75 


0.93 
3.18 

2.12 

1.68 
5-25 
2-43 
0-43 
13-50 


302 


STEEL  IN  TENSION. 


[Art.  34. 


TABLE   VIII.— Continued. 


PRODUCERS. 

ULTIMATE   RESISTANCE 
PER  SQ.  INCH. 

FINAL  STRAIN  OR 
ELONGATION. 

H.  Bessemer. 
Hard  Bessemer  steel  

Pounds. 

IO7  08^ 

Per  cent. 

I    87 

88  17^ 

J.  .O/ 

2O  OO 

78  606 

IQ    12 

Sanderson  Brothers. 

Cast  steel  from  Russia  iron  for  welding  .  . 
Double  shear  steel  

83,484 

IO7  Q4.O 

2.25 
o    qi 

Single  shear  steel               .  .               . 

IO7  182 

2   8l 

Fagot  steel  welded  .  .                     

•re  TOO 

I    2^ 

Drawn  bar   not  welded  

/Dj^-yy 
103  060 

3       A/I 

Messrs.  Turton  &  Sons. 
Steel  for  cups  

IOO  I<^ 

2.  7^ 

drills  

8?  ^2 

1.  06 

Qe  -172 

I-  ^7 

turning  tools      .  .        

80  27^ 

O    12 

machinery  

102  om 

I    4.^ 

IO2  ^67 

I    62 

mint  dies 

IC)6  2^7 

2    87 

dies  ... 

87  A7I 

o  87 

taDS  .  . 

Q7  QQ4. 

1.87 

7-1  266 

o  8r 

The  "  Area  of  Fracture  Section  "  (Table  IX.)  is  the/^r  cent. 
of  original  sectional  area,  which,  multiplied  by  that  original 
area,  will  give  the  area  of  the  fractured  section.  The  per  cent, 
of  contraction  will  then  be  given  by  taking  the  difference  be- 
tween 100  and  the  number  expressing  the  "Area  of  Fracture 
Section:' 

The  experiments  of  which  the  results  are  given  in  Table 
IX.,  are  those  for  which  the  coefficients  of  elasticity  were  com- 
puted in  Eqs.  (2),  (3),  (4)  and  (5). 


Art.  34.] 


ENGLISH  BAR   SPECIMENS. 


303 


TABLE   IX. 
Bar  Specimens. — 1868  and  1870. 


NUMBER  AND  KIND  OF  SAMPLE 

s. 

LIM.  OF  ELAS. 
PER  SQ.  INCH. 

ULT.  RESIST. 
PER  SQ.  INCH. 

PER  CENT.  FINAL 
ELONGATION. 

AREA  OF  FRAC- 
TURE  SECTION. 

5,  Hammered,  tires  .... 
5,  Hammered,  axles.  .  .  . 
4,  Hammered,  rails,.  .  .  . 
4,  Rol'd  tires,  axles,  rails. 
5,  Hammered,  tires    ... 
4,  Hammered,  axles.  .  .  . 
i,  Hammered,  rails.  .  .  . 
i    Rolled,  axles  J 

Crucible.  Bessemer. 

Pounds. 
52,200 
49,000 
48,000 
43,200 
46,200 
57,300 
44,000 
42  ooo 

Pounds. 
78,600 
75,000 

74,500 
71,500 
79,500 
91,700 
85,400 
68  600 

II.  I 

12.  1 
12.8 
17-5 
9.17 
8.72 
2.96 

10  56 

55-5 

51-4 

52-3 
65.0 
62.5 
72.1 
96.9 

89  .  9 

3    Chisel                                 ] 

eg  2AO 

1  18  200 

50 

OJ.    ^ 

3   Samples 

57  1  2O 

I  I  d  7OO 

7-1 

go  o 

Tires         ...              . 

eg  24O 

O7  J.OO 

47 

04.  8 

Rods  

0 

60  5OO 

QO  700 

I    i 

ICO  O 

2    Samples 

3 

45  900 

9O  8OO 

41 

QC      Q 

3    Gun-barrels    •  .  .    .      . 

L  o 

37  600 

86  300 

8  o 

f\c  ,  7 

Hammered    

£ 

56  ooo 

83  ooo 

8  o 

98.5 

Hammered 

44  800 

2    Rods         .    . 

CQ  QOO 

ye.  4OO 

O   Q 

Q8.7 

2   Rolled  J 

4C   QOO 

67  100 

2.O 

Q7.  1 

3Fasr£roted 

<3 

A1  7OO 

7O  7OO 

II    I 

55  8 

3    Samples 

44  800 

76  600 

II    Q 

£4.4 

2    Samples             .... 

^ 

•JQ    2OO 

II  .  c 

80.8 

3j  Tires  and  axles  

1 

07  OOO 

-e.  400 

13.6 

58.8 

Some  of  the  earlier  experiments  of  Mr.  Kirkaldy  (1861),  on 
various  English  (also  Krupp's)  steels,  gave  the  following  ulti- 
mate resistances  in  pounds  per  square  inch  for  specimens  of 
small  area  of  cross  section  : 

Highest  (forged  cast  steel,  reheated  and  cooled  gradually).  =  148,300 
Lowest  (forged  puddled  steel) =    42,600 

Table  X.  contains  the  results  of  twelve  experiments  by  Mr. 
Kirkaldy  ("  Experimental  Enquiry,  etc.,  of  Fagersta  Steel," 
1873)  on  2-inch  square  hammered  bars  of  Fagersta  steel,  turned 


304 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE  X. 
Fagersta   Steel  Bars. 


POUNDS  OF  STRESS  PER  SQ.  INCH,  AT 

PER  CENT.   OF  FINAL 

MARK. 

Elastic  Limit. 

Ultimate  Resist. 

Contraction. 

Elongation. 

FRACTURE. 

CO 

CO 

O 

\& 

CO 

1.2 
1.2 

62,400  )    ci 
60,200  V*° 

81,952  )   xn 
81,424  I* 

3-23  }   ci 

1-7)  H 
1.4!  " 

Granular. 

1.2 

63,500)    H 

92,224  )    H 

3.23)    " 

2-2)  ^ 

0.9 

63,600)    ^ 
62,400  [  * 

CO 
H 

vO 

112,976)  o 

109,952    ^     M 

H 
H 

4.97)  o 
8-39^  || 

H 

3-7)  ^ 
7-9^  II 

Granular. 

0.9 

63,200)    H 

96,912)       || 

4-97)      .; 

3-6)   ff 

§ 

CO 

O 

CO 

vO 

0.6 

62,500)  co~ 

101,232  )  g" 

21.46  )  2" 

5-5)   ^ 

0.6 

53,200  >  "^ 

97,968  t  2 

10.08  j-  |( 

Granular. 

0.6 

58,600)  II 

108,696)  II 

fc?lj 

CO  CO  CO 

odd 

8 

44,200)  co 

41,500  v  J 

40,600  )  II 

61,288  )  s 
63,120  v° 
59.528  )  II 

%•**}& 

60.42  V  .. 

62.61)  'I 

in 
22  .  1  j  "2 

Silky. 

> 

> 

^ 

^) 

< 

to  1.128  inches  diameter,  with  a  length  between  shoulders  of 
nine  diameters  or  10.15  inches. 


Art.  34.]          WHITWORTH'S  COMPRESSED   STEEL. 


305 


"Mark"  indicates  the  relative  hardness,  1.2  being  the 
hardest  and  0.3  the  softest. 

The  per  cents  are  of  the  original  sectional  area  (i.e.,  of  one 
square  inch)  and  of  the  original  length,  which,  of  course,  in- 
cludes the  section  of  failure. 

Sir  Joseph  Whitworth  manufactures  his  compressed  steel 
by  subjecting  the  molten  metal  to  an  intensity  of  pressure  of 
13,000  to  14,000  pounds  per  square  inch,  immediately  after  it 
is  taken  from  the  furnace. 

Table  XI.  contains  the  result  of  some  tensile  experiments 
on  some  specimens  of  this  steel.  Each  specimen  is  turned  to 
a  diameter  of  0.798  inch  (0.5  square  inch  in  normal  section)  for 
a  length  of  two  inches,  for  which  the  per  cent,  of  final  elonga- 
tion is  expressed  (see  "  Proc.  Inst.  of  Mech.  Engrs.,"  1875). 
The  specimens  are  thus  seen  to  be  so  formed  as  to  give  very 
high  results,  both  for  ultimate  resistance  and  elongation. 


TABLE   XI. 
Whitworth's    Compressed  Steel. 


DISTINGUISHING  COLORS  FOR 
CROUPS. 

ULT.  RESIST.  LBS. 
PER  SQUARE  INCH. 

PER  CENT.  FINAL 
ELONGATION. 

REMARKS. 

8O.6OO 

•12.  O 

(  Axles,  boilers,  cranks, 
\      propeller   shafts, 

Blue,  Nos.  1,2   3  

IO7  5OO 

24.  0 

(      rivets,  etc. 
\  Shafting,     drill    spin- 

Brown   Nos.  123.      .  . 

I2Q  QOO 

17.  0 

\      dies,  hammers,  etc. 
Large   planing    tools, 
Large  shears,  drills, 

Yellow,  Nos.  i,  2,  3  

Special  alloy  with  Tung- 
sten   

152,300 

161  300 

10.  0 
14..  O 

etc. 
Boring  tools,  finishing 
tools    for     planing, 
etc. 

20 


306 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE    XII. 
Plates. —  Unannealed. 


w 

TENSILE  STRESS   IN   POUNDS   PER  SQUARE 

h               z' 

o 

• 

§ 

§ 

O       H 

o 

INCH  AT 

U                h 

CHARACTER   OF 

i  s 

w 

«       O        5 

i           U 

I 

B"                       0 

r  KAC1  UKc.. 

B 

5 

Elastic  Limit. 

Ultimate. 

S       « 

o 

o 

0.30 

a 

43,260  )    « 

79,120  )     P, 

0.30 

2 

44,820  f     4 

77,840    f     «T 

0.30 

*(U 

45,no  I    ^ 

78,390  v  „ 

19-3 

Fine  and  silky. 

0.30 

s  1 

43,99° 

77,970 

0.30 

0  1 

44,720  J    ^ 

78,280  j  > 

O       p 

•S  vS 

Cd       c/5 

^  £ 

0.40 

O     £ 

51,620  )     §• 

81,990  )  <%. 

0.40 

"""'    <u 

50,980 

81,720  [  « 

0.40 

8  8 

51-260  V 

83,730  y  " 

13-9 

Very  fine. 

0.40 
0.40 

S  J 

51,100  I 
50,890  J     > 

81,830  i 
83,130  J  ^> 

41 

c    o 

••-i    vn 

0.50 
0.50 
0.50 

•°   "g 

58,950)  i 

59,200        co 

58,540  V  . 

85,790    )       Is 
86,22O      [          IT) 

85,560  L  " 

10.5 

Good  ;  slightly 
granular  on 

0.50 

0 

58,880      II 

86,000         H 

edges. 

0.50 

r.isj 

59,330  J  j> 

86,330  j     >* 

Boiler  Plate. 

Table  XII.  was  also  taken  from  Mr.  Hill's  paper,  and  con- 
tains results  obtained  from  tests  of  large  pieces  of  boiler  plate 
of  mild  steel  of  the  same  character  as  that  used  in  the  bars  the 
results  of  the  tests  of  which  are  given  in  Table  VII.  The  stress 
was  in  the  direction  of  rolling. 


Art.  34.] 


BOILER   PLATE. 


307 


TABLE  XIII. 
Plates. —  Unannealed. 


MEAN  OF 

DIMENSIONS    OF 
SPECIMENS  IN 
INCHES. 

STRESS  IN  POUNDS  PER  SQ.  INCH. 

PER   CENT.    OF 

STRETCH  OF 
ORIG'L  LENGTH. 

0    0 
u->   vr> 

m   n 

ocT    o" 
I^  O 

II      II 

j« 

Elastic  Limit. 

Ultimate. 

3 

If    X     4 

47,680 

66,440 

19.6" 

3 

4     x  i 

53,400 

68,900 

19.9 

<f> 

'C  - 

3 

2        X    | 

39,670 

62,360 

26.3 

J 

g, 

3 

3*  x  i 

38,940 

62,920 

27.7 

y 

Q 

8 

4 

2f    X    i 

38,390 

64,860 

24.8 

S 

3 

if  x   4 

53,790 

69,560 

ig.O 

o 

M 

1  : 

3 

4x4 

44,050 

65,830 

26.3 

* 

3 

2       X    f 

40,160 

65,690 

23-9 

to 

*rt   - 

3 

Si  x  I 

41,690 

65,200 

21-  5J 

C   - 

2 

I        X    4 

49,485 

77,270 

23.6 

rf 

+1 

2 

I     x  i 

50,745 

76,925 

24-8 

y 

.c 

•jjj   - 

2 

I        X    | 

42,985 

70,375 

29.8 

o 

>•£ 

2   " 

2 

I     x  | 

43,160 

70,530 

22.5 

*^ 

•M 

9 

i    x  4 

45,280 

64,970 

29.1 

«H 

*3    " 

7 

I        X     I 

41,810 

64,640 

28.6^ 

£ 

to 

5 

2       X    | 

41,368 

67,130 



S    to 

4 

2       X    | 

38,390 

66,990 



a  S 

13 

it  x  f 

39.430 

65,700 

For  10  ins.  23.2 

O  h-l 

Table  XIII.  contains  the  mean  results  of  Prof.  Kennedy's 
experiments  on  mild  boiler  plate  containing  about  0.18  per 
cent,  of  carbon  (London  "  Engineering,"  Vol.  XXXI.,  1881). 
The  left  column  shows  the  number  of  tests  in  each  group,  for 
which  the  other  columns  contain  the  mean  results.  The  dimen- 
sions of  specimens  are  not  exact,  but  closely  approximate.  The 
length  for  which  the  per  cent,  of  elongation  is  given,  in  all 
cases,  contained  the  fracture.  Consequently  those  "Per  cents" 
include  the  "  local "  extension  which  exists  at  the  section  of 
fracture.  This  accounts  for  the  larger  values,  as  a  rule,  which 
are  found  for  the  4-inch  lengths;  2^-inch  lengths,  containing 
the  section  of  fracture,  gave  much  higher  values. 


308 


STEEL  IN  TENSION. 


[Art.  34. 


The  first  five  groups  were  pulled  from  pins,  and  the  next 
four  from  wedge  grips.  The  manner  of  holding  the  test 
pieces,  however,  was  not  observed  to  have  any  influence  on 
the  results. 

Within  the  limits  of  these  experiments,  also,  the  ratio  of 
width  to  thickness  of  the  specimen  seemed  to  have  no  influ- 
ence. It  will  be  observed  that  Prof.  Kennedy's  specimens 
were  all  (what  may  be  called)  "  long  "  specimens. 

His  experiments  on  some  annealed  specimens  of  this  steel 
showed  that  the  process  of  annealing  reduced  the  ultimate 
resistance  only  3  or  4  per  cent. 

TABLE  xrv. 

Pagers ta   Plate. 


THICKNESS, 
INCH. 

ELASTIC  LIMIT  IN   LBS.   PER  SQUARE  INCH. 

ULT.   RESIST.   IN   LBS.    PER   SQUARE   INCH. 

Large. 

Small. 

Long. 

Mean. 

Large. 

Small. 

Long. 

Mean. 

Ill 

53,300 
37,900 
29,500 
31,100 
28,000 

50,500 
35,400 
29,300 
30,800 
28,300 

38,900 
35,600 
25,400 
27,500 
26,100 

47,567 
36,300 
28,067 
29,800 

27,467 

74,915 
60,480 
51,456 
55,803 
52,924 

71,940 
56,740 
50,345 
54,425 
52,475 

55,135 
54,  HO 
48,925 
50,160 
49,280 

67,330 
57,120 

50,243 

53,463 
51,560 

Mean. 

35,960 

34,860 

30,700 

33,840 

59>116 

57,185 

55,459 

52,715 
50,350 
50,842 
50,025 



51,878 

51,528 

55,943 

if} 

11! 

35,500 
33,800 
28,900 
27,800 
25,500 

33,200 
30,500 
28,100 
27,900 
25,700 

26,700 
29,800 
25,900 
27,300 
25,200 

31,800 
31,367 
27,633 
27,667 

25,467 

57,485 
54,543 
51,076 
51,338 
50,432 

45,460 
49.605 
46,740 
49,490 

47,455 

52,801 
52,288 
49,389 
50,557 
49,304 

Mean. 

30,300 

29,080 

26,980 

28,787 

52,975 

47,750 

50,868 

Fractures  all  "  silky." 


Art.  340 


BOILER  PLATE. 


309 


TABLE  XV. 

Fagersta  Plate. 


THICKNESS, 
INCH. 

PER   CENT.    CONTRACTION    OF  AREA. 

PER   CENT.   FINAL  STRAIN   OR  STRETCH. 

Large. 

Small. 

Long. 

Mean. 

Large. 

Small. 

Long. 

Mean. 

Unannealed. 

oo  px  Kf-  »|w^t-  «H 

43-1 
48.5 
59-3 
50.0 

55-1 

47.1 
54-2 
62.5 
58.6 
61.7 

37-9 
59-7 
71.0 
61.2 

60.7 

42.7 
54-1 

5^'  6 
59-2 

10.8 
28.2 
36.1 
36.4 

37-2 

13-5 
35-5 
4i.5 
40.0 

44-7 

5-21 
10.17 
20.64 
16.30 
'7-95 

9-45 
24.41 

32.57 
30.78 
33.01 

Mean. 

51.2 

56.8 

58.1 

55-4 

29.7 

'  35-0 

14.05 

26.04 

Annealed. 

oc|»ieK-*l«*-l—  *H 

57.1 
60.9 

63.4 
61.0 
62.0 

60.8 

63..  5 
63.6 
65.1 
64-3 

64.6 

67-5 
69.6 

64-3 
63-1 

60.8 
64.0 
65.5 
63-5 
63.1 

22.9 
33-8 
35.8 
38-5 
34-4 

28.4 
40.1 
42.0 
42.5 
43-5 

10.98 
16.88 
18.19 

17-45 

20.37 
29.99 
31.76 
33-08 
31-51 

29.52 

Mean. 

60.9 

63.5 

65.8 

63.4 

33-1 

39-3 

16.50 

In  his  paper  Prof.  Kennedy  explains  in  detail  his  "  elastic 
limit."  It  is  the  point  at  which  the  material  "  breaks  down," 
and  considerably  above  the  elastic  limit  as  analytically  defined 
in  this  work. 

Tables  XIV.  and  XV.  exhibit  the  results  of  Mr.  Kirkaldy's 
experiments  (in  the  direction  of  rolling)  on  some  Fagersta 

steel  plate  specimens. 
The  plate  from  which 

plate  specimen  these     specimens    were 

taken  was  marked  0.15, 
and  the  material  was 
a  m  il  d  steel.  The 


"Large"  and 


Fig.1 
"  Small 


specimens  were  shaped  as  shown  in 


310  STEEL  IN   TENSION.  [Art.  34. 

Fig.  I.  The  width  BC  or  AD  of  the  reduced  portion  was  ten 
inches  for  the  "  Large  "  pieces,  and  one  and  one  half  inches 
for  the  "Small"  ones.  For  the  "Large"  specimens,  the 
length  of  the  reduced  portion  (AB  or  CD)  was  ten  inches 
(  =  width),  and  four  and  one  half  inches  (  =  3  widths)  for  the 
"Small"  The  "Long"  specimens  were  100  inches  by  2j{ 
inches  "  in  the  clear." 

'The  results  embodied  in  these  two  tables  are  of  greater 
interest  and  value  in  consequence  of  the  variety  in  the  relative 
dimensions  of  the  specimens.  They  show  the  important  part 
played  by  "  lateral  strains  "  both  in  the  ultimate  resistance  and 
final  strains,  or  elongations,  of  test  specimens. 

With  very  few  exceptions  the  following  general  principle 
may  be  deduced  from  Table  XIV. : 

Both  the  elastic  limit  and  ultimate  resistance  increase  with 
the  ratio  of  the  width  over  the  thickness  of  the  plates. 

Nearly  all  the  exceptions  are  in  the  results  which  belong  to 
the  y%  unannealed,  and  the  "  Long"  annealed,  specimens.  It 
may  be  observed  in  connection  with  Table  III.,  that  the  char- 
acter of  the  former  specimen  (possessing  a  low  and  irregular 
value  of  E)  is  decidedly  abnormal,  to  which,  undoubtedly,  its 
exceptions  are  due.  Annealing  the  long  specimens  seems  to 
cause  the  disappearance  of  essentially  all  influence  of  the  rela- 
tive dimensions  of  the  cross  section,  where  the  ratio  of  width 
over  thickness  is,  comparatively  speaking,  small. 

One  origin,  of  the  results  above  stated  is  plainly  to  be 
found  in  the  lack  of  lateral  contraction  in  the  plane  of  the 
plate,  in  accordance  with  the  principles  shown  in  Article  32, 
"  Ultimate  Resistance  and  Elastic  Limit" 

An  examination  of  Table  XV.  shows  the  following  general 
result,  which,  however,  has  more  exceptions  than  the  pre- 
ceding : 

The  final  contraction  and  elongation  increase  as  the  ratio  of 
width  over  thickness  decreases. 

With  the  long  specimens,  this  does  not  seem  to  hold  for 


Art.  34.] 


BOILER  PLATE. 


less  values  of  the  ratio  than  21/ 


=  6.     Whether  these 


principles  may  hold  true,  as  general  ones,  or  whether  they  may 
hold  within  certain  limits  (a  possibility  indicated  in  the  "  Long" 
specimens),  the  number  and  character  of  these  experiments 
does  not  permit  to  be  decided.  They  show,  however,  that  the 
partial  prevention  of  lateral  strains  in  one  direction,  whatever 
may  be  the  cause,  will  affect,  to  a  considerable  extent,  experi- 
mental results  ;  also,  that  in  testing  plates  the  shape  and  rela- 
tive dimensions  of  the  test  piece  should  be  carefully  noted. 

TABLE  XVI. 
Open-Hearth  Steel  Plates— 1880. 


LENGTHWISE. 

CROSSWISE. 

SPECIMEN, 

PER  CENT. 

Stress  in  pounds  per 

1 

Stress  in  pounds  per 

•a 

c 

INCHES. 

square  inch. 

.  f- 

square  inch. 

««  xj 

CARBON. 

$  F 

c    u 

o     C 

Elas.  Lira. 

Ult.  Resist. 

&  " 

Elas.  Lim. 

Ult.  Resist. 

r 

3     X     14     X     18 

O.3O 

49,353 

93,339 

16 

49.510 

95-453 

18 

$  x   i£  x   15 

O.4O 

63,227 

86,410 

14 

63,723 

87,780 

16 

-ft-  x  i    x  12 

0.50 

65,070 

83,190 

10 

65,300 

84995 

15 

In  Table  XVI.  are  found  the  results  of  tests  by  Mr.  Hill 
("  Engineers'  Society  of  Western  Pennsylvania,"  April  2Oth, 
1880),  on  specimens  of  open-hearth  steel  plate.  Each  result  is 
a  mean  of  three,  and  each  specimen  was  cut  from  unannealed 
plate  in  a  planer.  It  is  to  be  particularly  observed  that  each 
thickness  of  plate  gave  essentially  the  same  elastic  limit  and  ulti- 
mate resistance,  whether  the  direction  of  the  testing  stress  was 
along  or  across  the  direction  of  rolling. 

Although  the  elastic  limit  increases  with  the  amount  of 


312 


STEEL  IN   TENSION. 


[Art.  34. 


carbon  (consistently  with  the  results  in  Table  XII.),  yet,  it  is 
very  remarkable  to  observe  that  the  ultimate  resistance  decreases 
as  the  carbon  increases,  which  is  not  consistent  with  the  results 
contained  in  Table  XII. 

TABLE   XVII. 

Siemens  Steel  Plate— 1875. 


THICKNESS 

POUNDS    OF   STRESS   PER   SQ.   IN.    AT 

PER  CENT. 

'PER  CENT. 

INCHES. 

Elas.  Limit. 

Ult.  Resist. 

STRETCH. 

TRACTION. 

e'O 

0-37 

34,600 

72,900 

22.3 

37-5 

5§ 

0.71 

30,400 

66,900 

24-5 

44-7 

3 

0-37 

31,500 

67,500 

24.8 

43-i 

1 

3 

0.40 

31,200 

66,400 

21    I 

44-7 

a 

1 

O.4O 

29,800 

66,100 

24.8 

38.5 

a 

O.5O 

29,400 

65,800 

26.4 

44-5 

3 

0.62 

26,300 

61,800 

25-5 

43-3 

0.70 

24,500 

60',  100 

25.0 

45-5 

ITJ 
.   §.2 

0.37 

34,300 

72,700 

22-4 

37-5 

G  rt 

PS 

0.71 

30,400 

67,300 

24.7 

43-6 

g 
g 

0-37 

31,200 

66,900 

26.4 

46.6 

I 

•H 

0.40 

31,000 

66,900 

26.3 

49.6 

u 

i 

0.42 
0.52 

30,000 
29,800 

65,800 

66,600 

2O-4 
20.2 

39-o 
46.7 

•< 

0.62 

26,300 

60,600 

22.7 

35-3 

0.70 

24,500 

60,  2OO 

26.O 

50.7 

The  ratio  of  width  over  thickness  of  specimen  increases 
from  2  (for  the  ^-inch,  or,  0.30  per  cent,  carbon)  to  5  ^  (for 
the  -j^-inch,  or,  0.50  per  cent,  carbon),  and  Mr.  Hill  considers 
this  an  explanation  of  this  disagreement  in  the  two  sets  of 
results.  The  results  of  a  large  number  of  tests  on  Fagersta 
steel  specimens  of  considerable  variety  in  the  ratio  of  width 


Art.  34.]  BOILER  PLATE.  313 

over  thickness  (Table  XIV.)  showed  a  regular  increase,  in 
both  elastic  limit  and  ultimate  resistance,  with  an  increased 
ratio  of  width  over  thickness.  Agreeably  to  these  results, 
therefore,  the  increase  of  carbon,  in  Mr.  Hill's  experiments, 
should  have  been  accompanied  by  an  increase  in  both  elastic 
limit  and  ultimate 'resistance,  since  an  increased  ratio '  of  width 
over  thickness  accompanied  the  increase  of  carbon.  The  dis- 
agreement seems  inexplicable,  but  was  probably  due  to  the 
influence  of  some  unnoticed  peculiarity  in  the  treatment  of 
the  material  in  the  original  plate,  or  of  the  specimens  them- 
selves. 

Table  XVII.  contains  the  results  of  some  specimen  tests  of 
Siemens  steel  plate,  made  by  Mr.  David  Kirkaldy  in  1875. 
The  per  cents  of  final  stretch  are  for  a  length  of  eight  inches, 
which  contained  the  section  of  fracture. 

Tables  XIII.,  XIV.,  and  XVII.  show  that,  as  a  general 
rule,  both  the  elastic  limit  and  ultimate  resistance,  in  mild  steel 
plates,  increase  as  the  thickness  of  the  plate  decreases. 

It  is  also  seen  that  the  process  of  annealing  decreases  both 
those  quantities. 

Although  Table  XVII.  shows  no  very  marked  result  in 
regard  to  final  stretch  and  contraction,  yet  when  it  is  taken  in 
connection  with  Table  XV.,  it  is  clear  that  the  process  of 
annealing  considerably  increases  both  the  final  stretch  and  con- 
traction ;  in  other  words,  increases  the  ductility  of  the  ma- 
terial. 

Again,  Table  XVII.  shows  that  the  ultimate  resistance  of 
steel  plates  is  essentially  the  same,  both  in  the  direction  of 
rolling  and  across  it.  This  result  is  in  agreement  with  that  of 
Mr.  Hill's  experiments,  as  well  as  that  of  French  experiments 
on  Bessemer  and  Martin  steel  plates  (Barba,  on  the  "  Use  of 
Steel,"  translated  by  A.  L.  Holley,  pages  26  and  29). 


STEEL  IN   TENSION. 


[Art.  34. 


Effects  of  Hardening  and  Tempering  Steel  Plates. 

In  connection  with  the  results  given  in  Table  XVII. ,  Mr. 
Kirkaldy  found  the  following  quantities  by  testing  the  same 
sized  specimens  of  the  same  plates  : 


Annealed. 


THICKNESS. 

0.64  inch. 
0.62  inch. 


ULTIMATE 
RESIST. 


FINAL 
STRETCH. 


.    57>IO°  pounds. . . . 
.    60,500  pounds. . . . 


24.1  per  cent. 

20. 2  per  cent. 


FINAL 
CONTRACTION. 

52.5  per  cent. 
48.7  per  cent. 


Hardened. 

0.64  inch....    64, 700  pounds ....    22.4  per  cent. ...    49. 3  per  cent. 
0.62  inch. ...    65,050  pounds. ...    18.0  per  cent. ...    45. 5  per  cent. 

The  hardening  was  done  by  heating   to  a  cherry-red   and 
cooling  in  water  at  a  temperature  of  82°  Fahr. 

TABLE    XVIII. 
Specimen   Tests. 


NOT  TEMPERED. 


TEMPERED. 


Pounds  of  stress  per  sq.  in.  at 

Per  cent. 

Pounds  of  stress  per  sq.  in.  at 

Per  cent. 

F*ina1 

Final 

Elas.  Lim. 

Ult.  Resist. 

Stretch. 

Elas.  Lim. 

Ult.  Resist. 

Stretch. 

58,350 

110,340 

13 

107,650 

169,430 

3-3 

56,800 

106,380 

15 

107,230 

163,320 

5-7 

55,720 

101,950 

17 

100,820 

153,370 

7-8 

53,178 

82,660 

19 

92,870 

140,580 

IO.2 

50,850 

91,460 

21 

88,170 

129,250 

12.6 

47,980 

84,780 

23 

78,110 

116,460 

14.8 

45,l6o 

78,110 

25 

70,700 

104,810 

17.0 

42,020 

71,700 

27 

63,480 

93,430 

19-5 

39,040 

66,300 

29 

56,800 

83,510 

22.  0 

33,510 

58,640 

32 

46,860 

73,560 

24.2 

Art.  34.]  RIVET  STEEL.  315 

Table  XVIII.  gives  the  results  of  some  experiments  on  a 
grade  of  French  mild  steel  known  as  me'tal  fondu.  It  is  made 
by  the  Bessemer  process,  or  in  a  Siejnens-Martin  furnace  by 
"  the  substitution  of  ferro-manganese  for  spiegel,  to  produce 
carbonization."  The  table  is  a  part  of  one  given  in  A.  L. 
Holley's  translation  of  "  The  Use  of  Steel,"  by  J.  H.  Barba. 
This  steel  was  produced  by  the  Siemens-Martin  process,  and 
the  specimens  were  small  ones,  turned  to  a  cross-sectional 
area  of  0.31  square  inch  throughout  a  length  of  3.93  inches. 

The  tempering  was  done  in  oil  at  a  bright  red  heat. 

It  is  thus  seen  that  tempering  or  hardening  increases  both 
the  elastic  limit  and  ultimate  resistance,  but  decreases  both 
the  final  stretch  and  contraction  of  area. 

In  these  French  experiments  "  it  was  observed  that  anneal- 
ing, well  done,  restored  to  the  metal,  in  every  case,  its  previous 
tenacity  and  elasticity." 

Rivet  Steel. 

In  Table  XIX.  will  be  found  the  results  of  the  experiments 
of  Prof.  Alex.  B.  W.  Kennedy  ("Engineering,"  6th  May, 
1881).  The  steel  was  a  very  mild  grade,  for  which  coefficients 
of  elasticity  have  already  been  given. 

The  specimens  were  turned  down,  as  shown,  from  •}-£,  if 
and  1-j1^-  inch  "rounds." 

As  in  all  other  cases,  the  elastic  limit  and  ultimate  resist- 
ance are  given  per  square  inch  of  original  section. 

Effect  of  Reduction  of  Sectional  Area,   in  connection  with 
Hammering  and  Rolling. 

Tables  XX.  and  XXI.  give  the  results  of  some  of  the  ex- 
periments of  Mr.  Kirkaldy  on  Fagersta  steel  bars.  The  bars 
were  originally  three  inches  square,  in  normal  cross  section, 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE    XIX. 
Rivet  Steel. 


POUNDS  OF  STRESS   PER  SQ.   IN.   AT 

PER    CENT.    OF 

ORIGINAL  DIA.   OF 
BAR. 

DIAMETER    OF 
SPECIMEN. 

FINAL  STRETCH 
IN   10  INCHES. 

Elas.  Limit. 

Ult.  Resist. 

Inch. 

Inch. 

Q 

0 

1 

0.512 
0.507 

43,400)    3 
45,2ooV  J 

45,73o)    II 

64,770)  g 

65,500^  7J 
65,770)    'I 

\      O 
22.5!     « 

'            > 

! 

0.616 
0.622 
0.616 

46,370)  o; 
46,200  V  |J 
46,220  )  il 

o 

CO 

oo 
67,960)   oo" 

69,310^ 

69,210)  11 

0 

19.2)  S 

21-3  f    || 

19.1  )  ^ 

f 

o 

CO 

il 

0.804 
0.804 
0.786 

48,600)    rC 
47,  730  ^    n 
46,750)     l[ 

60,280)    t-T 

60,750^ 

63,500)  • 

21.6  )    S? 
22.2  >•    || 
26.0) 

and  were  hammered  or  rolled  down  to  the  dimensions  shown 
in  the  second  column  in  each  table.  Specimens  were  then 
turned  down  for  testing  to  the  diameters  given  in  the  third 
column,  for  a  length  of  ten  inches.  The  tables  give  results  for 
duplicate  specimens,  one  set  having  been  unannealed  and  the 
other  annealed.  The  fractures  belonging  to  the  3x3  bars 
were  all  granular,  and  those  belonging  to  the  0.5  x  0.5  bars 


Art.  34.] 


HAMMERING  AND  ROLLING. 


317 


TABLE  XX. 

Fagersta  Steel. —  Unannealed. 


BARS  IN  INCHES. 

DIA.    OF 

SPECIMENS, 

POUNDS  OF 
SQUARE 

STRESS   PER 
INCH  AT 

PER  CENT. 
FINAL 

PER  CENT. 
FINAL 

INCHES. 

Elas.  Limit. 

Ult.  Resist. 

STRETCH. 

CONT. 

Hammered.  . 
Rolled  

0.5  x  0.5 

O.  <    X    O    5 

0-357 

O.^*7 

78,300 
4.6  800 

95,960 

QO  7^O 

6.9 

16  o 

47.0 

4,-j.O 

Hammered.  . 
Rolled 

I    X     I 
I    X    I 

O.6I9 
O   6lQ 

49,800 

A"!    IOO 

83,720 

87  760 

16.0 
16.2 

44-7 

2Q.'J 

Hammered.  . 
Rolled 

1.5  x  1-5 

I    c     x    I    f> 

1.009 

I    009 

46,700 

JO  ^OO 

77,720 

7O  280 

12.6 
IO   2 

38.8 

is  8 

Hammered.  . 
Rolled  

2X2 
2X2 

1.382 
1.382 

44,800 

38,300 

80,920 

84,073 

19.2 
15.9 

35.5 

20.8 

Hammered.  . 
Rolled.  . 

2.5   x  2.5 

2    C    x    2    H 

1.694 

I    604 

34,700 

36  600 

78,840 

72  efts 

21.4 
8   2 

26.2 

IO    ^ 

Hammered.  . 
Rolled  .  . 

3><3 
3X    "\ 

1.994 

I    OQ4 

38,800 

30  400 

70,080 
62  ^o1? 

2.3 

2e 

4.4 

4      A 

all  silky ;  the  intermediate  ones  were  partially  silky  and  par- 
tially granular. 

As  a  part  of  the  hammering  and  rolling  was  done  at  such  a 
temperature  as  to  essentially  amount  to  cold  hammering  or 
cold   rolling,  the    annealed    specimens   show    more   truly  the 
effects  of  the  two  kinds  of  treatment  than  the  others. 
The  following  results  can  be  at  once  observed : 
The  elastic  limit,  ultimate  resistance  and  final  contraction  at 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE   XXI. 

Fagersta   Steel. — Annealed. 


BARS  IN  INCHES. 

DIA.    OF 
SPECIMENS, 

POUNDS   OF 
SQUARE 

STRESS   PER 
INCH  AT 

PER  CENT. 
FINAL 

PER  CENT. 
FINAL 

INCHES. 

Elas.  Limit. 

Ult.  Resist. 

STRETCH. 

CONT. 

Hammered  .  . 
Rolled 

0.5   x  0.5 

o  x  x  o  ^ 

0-357 
O   ^7 

47,800 
41  2OO 

82,120 

So  2IO 

7-7 
o  8 

55-0   . 
m  o 

Hammered.  . 
Rolled  

I     X     I 
I     X     I 

O.6ig 
0.619 

40,800 

40  loo 

78,650 

83,720 

15.2 

II  .^ 

54-o 

3Q-  7 

Hammered.  . 
Rolled  

1-5   x   1-5 

I  .  ^    X    I  .  % 

1.009 
I  .OOQ 

42,300 

37,800 

77,810 

82,780 

13-7 

m.2 

47-7 
38.7 

Hammered.  . 
Rolled 

2X2 
2X2 

1.382 
I    ^82 

41,300 

•16,100 

78,893 

80  ^o 

17.7 

16  8 

41.2 
38.0 

Hammered.  . 
Rolled  

2.5  x  2.5 

2.  5    X    2.  5 

1.694 
1  .604 

31,300 

32,700 

66,140 
71,630 

14.7 
13.8 

45-4 

35-  5 

Hammered  .  . 
Rolled 

3X3 
o    X    ^ 

1.994 
I    QQ4. 

29,800 
27,600 

69,640 

60  IQ3 

7-7 
3.8 

8.4 
e  .4. 

section  of  fracture  increase  very  much  with  the  decrease  of  sec- 
tional area  for  either  the  hammered  or  rolled  bars. 

Other  and  similar  experiments  verified  these  conclusions 
for  both  higher  and  milder  Fagersta  steels. 

The  per  cents,  of  final  stretch  are  the  greatest  for  the  inter- 
mediate sectional  areas,  whether  annealed  or  unannealed,  while 
the  relative  effects  of  rolling  and  hammering  are  irregular. 

The  hammered  specimens  invariably  give  the  greatest  final 
contraction,  whether  unannealed  or  annealed. 


Art.  34.]  WIRE.  319 

If  unannealed,  the  hammered  specimens  give  the  highest 
elastic  limit  and  ultimate  resistance ;  if  annealed,  while  this 
holds  true  (essentially)  for  the  elastic  limit,  the  rolled  specimens 
give  the  highest  ultimate  resistance  in  four  out  of  the  six  tests. 

Annealing  decreases  both  the  elastic  limit  and  ultimate  re- 
sistance ;  this  was  also  found  to  be  the  case  for  both  higher  and 
milder  Fagersta  steel  specimens,  which  were  similarly  tested. 

In  a  set  of  24  experiments  (precisely  the  duplicates  of 
those  whose  results  are  given  in  Tables  XX.  and  XXI.)  with  a 
higher  grade  of  steel,  the  greatest  final  stretch  was  found  to 
belong  to  the  smaller  cross  sections ;  while  in  a  similar  set  with 
a  milder  grade  of  metal,  the  greatest  final  stretch  was  found 
with  the  larger  bars,  whether  the  specimens  were  unannealed 
or  annealed. 

Other  relative  effects  of  hammering  and  rolling  were  some- 
what irregular,  and  seemed  to  depend  on  the  grade  of  steel. 

Effects  of  Annealing  Steel. 

It  has  not  been  convenient  to  separately  classify  the  experi- 
mental results  showing  the  effects  of  annealing,  but  it  has  been 
seen  that  the  process,  in  general,  decreases  both  the  elastic 
limit  and  ultimate  resistance,  and  increases  the  ductility ;  the 
lower  grades  of  steel  being  the  least  influenced. 

Steel  Wire. 

Table  XXII.  contains  the  results  of  testing,  to  ultimate 
resistance,  the  wire  for  which  the  coefficients  of  elasticity  were 
given  in  Table  I.,  together  with  some  belonging  to  the  Chrome 
Steel  Co.'s  wire,  also  tested  by  the  engineers  of  the  New  York 
and  Brooklyn  bridge.  The  diameter  of  this  wire  was  about 
0.165  inch  (No.  8  Birmingham,  gauge).  As  will  presently  be 
shown,  some  of  the  material  was  cast  steel  and  other  Bessemer 
steel,  all  having  been  hardened  and  tempered. 


320 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE    XXII. 
Steel  Wire. 


c/i 

i 

| 

ULTIMATE  RESISTANCE  IN 

E     s 
u 

^    5 

i 

POUNDS  PER  SQ.  IN. 

6'   S 

«     5 

Is.          ^ 

PRODUCER. 

0 

S    fe 

U 
<         K 

0 

Greatest. 

Mean. 

Least. 

E  1 

3     g 

J   Lloyd  Haigh                                   (i) 

12 

182,450 

175,340 

166,169 

4.9 

1-7 

o.  161 
0.147 

Cleveland  Rolling  Mills                    (2) 

6 

182,576 

178,400 

172,984 

4-2 

2.1 

0.161 
0.138 

Washburn  &  Moen                           (3) 

6 

184,019 

I?6  4^7 

169,706 

4-5 

0.147 

' 

0.8 

o.i33 

Sulzbacher,  Hymen,  Wolff  &  Co.  .(4) 

6 

179,833 

175,291 

167,807 

4-4 
1.8 

0.162 
0.139 

John  A.  Roebling's  Sons  Co  (5) 

'3 

179,019 

162,244 

125,321 

4.8 
0.4 

0.167 
0.130 

'Johnson  &  Nephew                           (6) 

9 

206,170 

177,706 

163,027 

6.9 

0.148 
0.129 

Carey  &  Moen  (7) 

12 

194,227 

167,880 

126,814 

4.2 
o-5 

0.160 
0.125 

Chrome  Steel  Co  (8) 

6 

170,150 

160,544 

^50,657 

3-4 

I.O 

The  column  "  Per  cent,  final  stretch "  gives  the  highest 
values  for  the  5-feet  lengths  tested,  and  the  lowest  for  the  100- 
feet  lengths ;  these  were  the  greatest  and  least  found. 

The  column  "  Dia.  fracture "  gives  the  greatest  and  least 
values  of  the  diameter  of  the  fractured  section  in  decimals  of 
an  inch.  There  seemed  to  be  no  definite  relation  between  the 
ultimate  resistance  and  contraction  of  section  of  rupture. 

Col.  W.  A.  Roebling  states  that  the  character  of  the  above 
steel  was  believed  to  be  as  follows : 

\       (i)  English  crucible  cast  steel. 

(2)  Open-hearth  steel. 

(3)  English  crucible  cast  steel. 

(4)  Krupp's  Bessemer  and  cast  steel. 

(5)  Crucible  cast  steel  and  American  Bessemer  steel. 
($)  English  crucible  cast  steel. 

(7)  English  crucible  cast  steel. 

(8)  Crucible  cast  steel.  —,.,m 


UNIVERSITY 

Art.  34.]  SHAPE   STEEL. 

In  Fairbairn's  "  Useful  Information  for  Engineers/'  3d 
series,  p.  282,  the  following  values  are  given  for  English  steel 
wire  (1866): 

ULT.   STRETCH 
PRODUCER.  .  DIA.  ULT.   RESIST.  FOR   50  INS. 

Jenkins  &  Hill,  soft  patent  steel  . . .  .0.085  in 105,730  Ibs 0.53 

Jenkins  &  Hill,  annealed  steel 0.085  in 79.297  Ibs 5. 50 

Johnson 0.095  in 275,100  Ibs 1.71 

Johnson,  patent  steel • 0.095  in 275,100  Ibs 1.26 

The  ultimate  resistance  is  in  pounds  per  square  inch,  and 
the  final  stretch  in  per  cent,  of  original  length  of  50  inches. 

It  is  therefore  seen  that  steel  drawn  into  wire  possesses  an 
excess  of  resistance  over  that  in  larger  masses,  as  bars  ;  it  thus 
exhibits  the  same  general  phenomenon  as  wrought  iron  under 
similar  circumstances.  In  fact  the  shape  and  dimensions  of 
specimens  have  been  seen  to  exhibit  the  same  general  effects 
on  the  results  of  testing  as  were  found  with  wrought  iron. 

Shape  Steel. 

The  following  mean  results,  found  by  testing  specimens  of 
Bessemer  and  Martin  Is  and  Ls>  are  given  in  A.  L.  Holley's 
"  Use  of  Steel,"  by  J.  Barba  : 

Untempcred  Specimens. 

ULT.   RESIST.  ULT.   STRETCH. 

Bessemer  Is 73, 500  Ibs.  per  square  inch. 19.5  per  cent. 

Bessemer  Is 74,790  Ibs.  per  square  inch 21 . 1  per  cent. 

Martin   Ls 64,960  Ibs.  per  square  inch 21.7  per  cent. 

Martin   Ls 67,210  Ibs.  per  square  inch 24.5  per  cent. 

Tempered  Specimens. 
Bessemer  Is 106^830  Ibs.  per  square  inch 6.4  per  cent. 

The  tempering  was  done  by  heating  to  cherry-red  and  cool- 
21 


322 


STEEL  IN   TENSION. 


[Art.  34. 


ing  in  water  at  50°  Fahr.     The  dimensions  of  the  specimens 
are  not  given. 

Steel  Gun   Wire. 

In  1875,  W.  E.  Woodbridge,  M.D.,  made  a  large  number  of 
tests  on  the  mechanical  properties  of  steel  gun  wires.  The 
"  wires  "  were  about  0.3  inch  square,  having  been  drawn  down 
from  bars  0.375  mcn  square.  The  full,  detailed  account  of 
these  experiments  is  given  in  "  Report  on  the  Mechanical 
Properties  of  Steel,  etc.,  by  W.  E.  Woodbridge,  M.D." 

The  results  given  in  this  section  are  abstracted  from  the 
"  Report  "  mentioned. 

TABLE   XXIII. 

Gun    Wires — Annealed. 


KIND  AND   MANUFACTURER. 

POUNDS   OF 
SQ.    ] 

STRESS   PER 
N.    AT 

ENT.  FINAL 
RETCH. 

ENT.  FINAL 
-RACTION. 

Elas.  Lim. 

Ult.  Resist. 

u    R 

tt 

m 

A< 

U      g 

B  8 

Crucible  steel  ;  Hussey,  Welles  &  Co  (10) 
«           «i             «             <>         < 

41,100 
26,800 

92,300 
50,700 

5-8 

22.  O 

45-0 
67.0 

It                         14                              «                              <f                    4 

^4,OOO 

61,700 

16.0 

«5Q.O 

«                         <(                              <«                              «                    < 

OQ  700 

71  600 

I^.O 

4O.  O 

Martin  steel  •  N   J    Steel  &  Iron  Co 

42  ^OO 

72,IOO 

17.3 

**?tW 

46.0 

1  (10) 
«          «          t«         (i           «       < 

37,500 

•3Q    2OO 

71,800 
71,  coo 

21.5 

iS.o 

46.0 
46.0 

«          «          «         «           «        < 

4Q  OOO 

Q4,  6OO 

14.1 

•57.0 

"  Gun-screw  wire  "  iron  ;  Trenton  Iron  Co.  .  . 
Chrome  steel  ;  Chrome  Steel  Co  (10) 

24,700 
A  7  4.00 

y^,  V.HJU 
52,600 
89,OOO 

21.  I 

9-  I 

57-0 
61.0 

<C                          «                                 «                            <t                1  « 

3Q.2OO 

77,7OO 

6-3 

6r.o 

«          «             «           «      «< 

43,200 

7I,IOO 

14.2 

44.0 

«               a                  «                ««         .<                          /IQ\ 

qn  100 

89,IOO 

8.7 

41.0 

Norway  iron  *  Messrs   Naylor  &  Co  . 

26,OOO 

47,8OO 

28.5 

70.0 

German  steel  ;  Messrs.  Park  Bros.  &  Co  .... 
Cemented  cast  steel  ;  Messrs.  Park  Bros.  &  Co. 

22,IOO 

35.100 
26,700 

51,700 
6l,7OO 

57,700 
74,900 

14.2 

15-7 
17.0 

19-5 

42.0 
52.0 
50.0 
33-0 

Art.  34.]  EFFECT  OF   TEMPERATURE.  323 

Table  XXIII.  gives  results  for  wires  which  were  annealed  at 
bright  red  heat,  without  oxidation. 

The  per  cents,  of  final  stretch  are  for  five  inches  of  original 
length,  except  in  the  case  of  specimens  marked  "(10),"  which 
indicates  that  the  per  cents,  are  for  ten  inches  of  original 
length. 

Other  tests  of  wires  about  0.3  inch  square  and  unannealed, 
gave  the  following  ultimate  resistances  in  pounds  per  square 
inch  of  original  section.  The  wires  were  of  different  varieties 
of  steel,  including  cast  and  Martin  steel. 

130, 800.  84,400. 

106,900.  58,700. 

108,200.  59,200. 
135,000. 

The  elastic  limit  varied  from  35  to  92  per  cent,  of  the  ulti- 
mate resistance ;  and  the  per  cent,  of  final  contraction  varied 
from  1 1  to  43.  The  effect  of  annealing,  both  on  resistance  and 
ductility,  is  made  very  evident  by  comparing  the  two  sets  of 
results. 


Effect  of  Low  and  High  Temperatures  on  Steel. 

The  results  of  some  German  experiments  and  the  expe- 
rience of  the  Massachusetts  Railroad  Commissioners  with  steel 
rails  for  one  year,  have  already  been  given  in  connection  with 
wrought  iron. 

Table  XXIV.  contains  the  results  of  the  experiments  by 
Mr.  Chas.  Huston,  as  given  in  the  "  Journal  of  the  Franklin 
Institute  "  for  Feb.,  1878. 

"  U.  R."  is  the  ultimate  resistance  in  pounds  per  square 
inch,  while  "  C."  is  the  per  cent,  of  contraction  at  the  section 
of  fracture. 

Each  result  is  a  mean  of  three  experiments. 


324 


STEEL   IN   TENSION. 


[Art.  34. 


TABLE    XXIV. 


KIND    OF   MATERIAL. 

"  COLD." 

572°    FAHR. 

932°    FAHR. 

U.  R. 

C. 

U.  R. 

C. 

U.  R. 

C. 

Charcoal  boiler-plate    piled.     

55,400 
54,600 
64,000 

78,400 

26 

47 
36 

27 

63,100 
66,100 
69,300 

82,800 

23 

.38 
30 

16 

65,300 

64,400 

68,600 

77,300 

21 

34 

20 

Siemens-Martin  (exceptionally  soft).  .  . 
Crucible  steel  (ordinavily  soft) 

Crucible  steel  (not  quite  hard  enough 
to  temper)              .        .... 

The  method  of  producing  rupture  at  the  desired  place  was 
such  as  to  make  the  specimens  partake,  to  some  extent  at 
least,  of  the  nature  of  "  short  "  ones,  which,  however,  would 
not  affect  the  comparative  results. 

It  will  be  observed  that  the  charcoal  boiler-plate  iron  gave 
the  highest  resistance  at  the  highest  temperature,  but  that  all 
the  steels  gave  the  highest  "  U.  R."  at  the  intermediate  tem- 
perature 572°  Fahr. 

It  is  somewhat  remarkable  that  in  every  case  but  the  last 
(the  hardest  steel) -the  contraction  of  fractured  section  de- 
creased with  the  rise  in  temperature. 

Other  results  for  steel  will  be  found  in  Table  IX.  of  Article 
35,  and  it  will  be  seen  that  they  tend  to  confirm  the  conclu- 
sions just  drawn. 

In  the"Annales  des  Fonts  et  Chaussees  "  for  Feb.,  1881, 
page  226,  are  given  the  number  of  breakages  of  steel  rails 
which  occurred  in  Russia  in  1879.  The  following  is  the  table 
showing  the  number  of  failures  for  each  month  of  the  year. 

These  results  conflict  somewhat  with  those  given  by  the 
Massachusetts  Railroad  Commissioners,  in  Art.  32. 


Art.  34.]  CONSTRUCTIVE   PROCESSES.  325 

January 699 

February 598 

March 854 

April 235 

May 235 

June 160 

July 247 

August 156 

September 214 

October 328 

November 341 

December 692 

The  greatest  number  is  found  in  the  coldest  half  of  the 
year,  but  the  greatest  number  for  any  one  month  belongs  to 
March,  which  is  not  the  coldest  month.  It  is  probable  that 
this  is  due  to  the  effect  of  long  wear  on  the  frozen  ground 
of  the  entire  winter  in  connection  with  the  possible  alternate 
freezing  and  thawing  of  the  ground  in  the  month  of  March. 

Effect  of  Manipulations  common  to  Constructive  Processes ;  also 
Punched,  Drilled,  and  Reamed  Holes. 

Table  XXV.  gives  the  results  of  the  experiments  of  Mr. 
Hill  (paper  already  cited  in  connection  with  Tables  V.  and 
VI.)  on  specimens  of  exactly  the  same  size,  and  from  the  same 
steel  plates,  as  those  given  in  Table  XVI. 

The  different  methods  of  preparing  and  treating  the  speci- 
mens are  shown  in  the  column  headed  "  Treatment  of  speci- 
men." 

With  the  exception  of  those  of  the  lowest  two  0.30  per 
cent,  specimens,  the  results  are  averages  of  a  number  of  ex- 
periments. 

From  these  results  Mr.   Hill    concludes:    1st.  "That  both    , 
shearing  and  punching  are  injurious,  per  sc,  to   all   grades  of 
steel,  and  cold  punching  far  more  so  than  shearing. 

2d.    "  That    both    these   operations    affect  the  elastic  limit  ; 
.     .     .     far  more  than  they  do  the  ultimate  resistance. 


326 


STEEL   IN   TENSION. 


[Art.  34. 


TABLE   XXV. 

Open-Hearth  Steel 


PER  CENT. 
OF 

TREATMENT   OF  SPECIMEN. 

POUNDS  OF 
SQUARE 

STRESS   PER 
INCH   AT 

PER   CENT. 
OF  FINAL 

CARBON. 

Elas.  Lim. 

Ult.  Resist. 

STRETCH. 

Cut  in  planer 

AQ  4^1 

Q/1      7  Q6 

17    OO 

O-7Q 

Sheared    .            .... 

•30  ^7O 

74  080 

jO   OO 

o  30 

Punched  

o 

6^  4IO 

O   45 

o  30 

Punched  and  hammered  cold      

o 

87,540 

O.  55 

0.30 

Punched,  hammered  and  annealed  .  . 

55,780 

100,410 

7-50 

o  40 

Cut  in  planer 

Ao  /17C 

87  OQ5 

15    OO 

o  40 

46  QOO 

75,330 

7.OO 

o  40 

Punched        .  . 

o 

68  890 

5  .OO 

o  40 

Sheared  and  annealed       

CQ    TCQ 

86  1  60 

16.00 

o  40 

CO    78O 

jo3  560 

7    OO 

o.  50 

Cut  in  planer  

65  185 

84.092 

I2.5O 

o  50 

Sheared 

51  666 

7Q  QOO 

5  .OO 

o  50 

Punched                         .            .... 

o 

78  400 

4.OO 

o  50 

Sheared  and  tempered  

60  375 

87,203 

I7.OO 

o  50 

57  060 

84,900 

12.  OO 

3d.  "  That  apparently  the  lower  grades  of  steel  are  pro- 
portionately more  injuriously  affected  than  the  higher 
grades.  .  .  . 

4th.  "  That  the  injurious  effects  of  shearing  and  punching 
can  be  almost  entirely  counteracted  by  subsequent  annealing, 
or  tempering  in  oil  from  a  low  heat. 

5th.  "  That  annealing  restores  the  elastic  limit  to  a  greater 
extent  than  the  ultimate,  while  tempering  as  above,  on  the 
contrary,  largely  increases  the  ultimate  resistance  and  ductility, 
but  does  not  so  fully  restore  the  elastic  limit." 

Table   XXVI.   shows  the  results  of  other  experiments  by 


Art.  34.] 


PUNCHING,   DRILLING,   ETC. 


327 


Mr.  Hill,  from  the  same  paper,  on  the  relative  effects  of  drill- 
ing, punching  and  reaming,  and  punching  (with  and  without 
annealing)  rivet  holes  in  steel  plates.  These  plates  were  pre- 
cisely the  same  as  those  from  which  the  results  given  in 
Tables  XVI.  and  XXV.  were  obtained.  The  dimensions  of 
the  different  specimens  are  given  in  the  second  column  from 
the  left. 

TABLE    XXVI. 

Open-Hearth  Steel. 


.PER  CENT.  CARBON. 

PLATE   SPECIMEN. 

HOLE. 

ULT.  RESIST.,  LBS.  PER 
SQ.  IN.  OF  EFFECTIVE 
SECTION. 

PER  CENT.  ELONGATION 
OF  HOLE. 

08  066 

K  in.  rolled  plate,  cut 
in  planer  on  all  edges. 

Punched,  o.Q35.in.  /  dj 

0.30 
0.30 

Strips  2^   ins.  wide,   18 
inches  long. 

Reamed  to  i.i  in.   f 
Punched  and  annealed,  0.935  in-  diameter  .  . 
Punched,  0.955  in.  diameter     

78,970 

66  108 

21.0 

Drilled,  0.6  in.  diameter   

Tr    g 

%  in.  rolled  plate,  cut 

Punched,  0.5  in.      {.diameter. 

0,40 

as  above.     Strips  i  V\  ins. 
wide,  15  inches  long. 

Reamed  to  0.62  in.  f 
Punched  and  annealed,  0.62  in.  diameter  ... 
Punched,  0.62  in.  diameter        .     ... 

87,910 

18.9 

Drilled  o  4  in  diameter         ...      .          .   .. 

86963 

2Q   O 

13b'  rolled  plate,  cut  as 

Punched,  0.4  in.     ldiameter< 

89,043 

26.0 

o.  50 

above.     Strips    i     inch 
wide  12  inches  long. 

Reamed  to  0.5  in.  j  u 
Punched  and  annealed,  0.45  in.  diameter  
Punched   o  45  in.  diameter 

84,95i 
82,330 

31.0 
15.0 

The  relative  influences  of  the  different  operations  of  drill- 
ing, punching,  etc.,  will  be  emphasized  by  comparing  the  re- 
sults in  Table  XXVI.  with  those  given  in  Table  XVI. 

The  operation  of  punching  is  seen  to  considerably  injure 
the  material  in  the  vicinity  of  the  punched  hole.  In  every 
case,  the  punched  specimen  gives  very  much  less  resistance 


328  STEEL   IN   TENSION.  [Art.  34. 

than  any  other.  It  is  further  to  be  observed  that  the  injurious 
effect  of  the  punch  is  only  partially  removed  by  annealing. 

Mr.  Hill  draws  the  following  conclusions  : 

1st.  "  That  the  '  reamed'  hole  is  the  strongest,  and  follow- 
ing in  the  order  of  strength  come  the  '  drilled/  the  '  punched 
and  annealed,'  and,  lastly,  the  '  cold  punched '  hole.  This 
graduation  is  well  defined  in  all  three  groups.  That  the 
reamed  hole  should  be  stronger  than  the  drilled  hole,  I  am 
unable  to  account  for. 

2d.  "  That  the  injurious  effect  of  punching  is  local,  and  can 
be  entirely  removed  by  enlarging  the  hole  sufficiently  with 
either  drill  or  reamer.  The  amount  of  drilling  or  reaming  re- 
quired after  punching  varies  with  the  thickness  of  the  plate 
and  grade  of  steel. 

3d.  "  That  although  annealing  is  in  a  measure  beneficial  in 
partially  restoring  strength  and  ductility  to  the  punched  plate, 
it  will  hardly  be  found  available  for'bridge  work  ;  for,  if  you 
attempt  to  anneal  before  riveting,  the  holes  will  not  fit  ;  if  after 
riveting,  you  create  internal  strains  of  which  no  account  can  be 
taken,  and  which  may  subsequently  produce  failure.  More- 
over, with  proper  machinery,  punching  and  reaming  will  be 
found  much  cheaper  than  '  punching  and  annealing.'  ' 

In  regard  to  the  excess  of  resistance  with  the  "punched  and 
reamed  "  hole  as  compared  with  the  drilled,  it  may  be  remarked 
that  in  every  case  the  hole,  as  reamed,  was  greater  in  diameter 
than  the  drilled  hole.  There  was,  consequently,  less  material 
to  be  ruptured  in  the  former  case  than  the  latter.  This 
diminution  of  cross  section  makes  the  reamed  specimen  a 
"smaller"  one  than  the  other  and  intensifies  the  "  shortening" 
effect  of  the  rivet  hole.  Both  these  influences  would  increase 
the  ultimate  resistance,  per  square  inch,  of  the  reamed  speci- 
men beyond  the  drilled  one  ;  but  whether  they  supply  the 
explanation  for  the  whole  difference  is  an  open  question. 

It  will  be  observed  that  the  ultimate  resistances  for  the 
drilled  and  reamed  specimens,  in  Table  XXVI. ,  run  consider- 


Art.  34.]  PUNCHING,   DRILLING,   ETC.  329 

ably  higher  than  the  corresponding  quantities  in  Table  XVI., 
or,  indeed,  those  in  Table  XXV. ;  for  which  the  explanation 
is  simple  and  obvious.  The  specimens  for  Table  XXVI.  car- 
ried one  rivet  hole  each  ;  and  at  this  rivet  hole  failure  took 
place.  The  effect  of  the  hole  in  any  specimen. was  the  restric- 
tion of  the  contraction  to  its  immediate  vicinity,  and  the  par- 
tial prevention  of  the  latter,  which  reduced  the  specimen,  to  a 
great  extent,  to  a  "short"  one.  An  increase  of  ultimate  re- 
sistance was,  consequently,  to  be  expected. 

The  decrease  of  ultimate  resistance  with  the  increase  of 
carbon  has  already  been  remarked  upon  in  connection  with 
Table  XVI. 

The  experiments  of  Prof.  Alex.  B.  W.  Kennedy,  on  the 
effect  of  punching  and  drilling 
holes,  in  mild-steel  boiler  plate, 
are  well  illustrated  by  Table 
XXVII.,  which  is  condensed 
from  one  given  in  London 
"Engineering,"  6th  May,  1881. 


o 


r\ 


None  of  these  plates  were  an- 
nealed, but   all  were  drilled   or  Fig.  2 
punched  as  received. 

Within  the  limits  of  these  experiments,  Prof.  Kennedy 
observes,  neither  the  width  of  the  test  piece  nor  the  different 
diameters  of  die,  had  any  essential  influence  on  the  results. 

The  injurious  effect  of  punching  is  shown  by  the  fact  that 
the  punched  specimens  gave  only  92  to  98  per  cent,  of  the 
resistance  of  the  drilled  ones. 

It  will  be  noticed  that  both  the  drilled  and  punched  speci- 
mens gave  higher  resistances  than  the  natural  plate.  This  is 
due  to  the  "  shortening  "  and  other  influence  (/.  r.,  the  disturb- 
ance of  the  lateral  strains)  of  the  rivet  holes,  as  before  ob- 
served, and  explained  in  Art.  32,  "  Ultimate  Resistance  and 
Elastic  Limit." 


330 


STEEL  IN   TENSION. 


[Art.  34. 


TABLE  XXVII. 

Punched  and  Drilled  Holes. 


THICKNESS  IN 
INCHES. 

HOLE,   INCHES. 

DIAMETER   OF   HOLE, 
INS. 

TENACITY  IN 
TONS,  SQ.  IN. 
NET  SECTION. 

TENACITY   COMPARED   WITH 
THAT   OF 

Na.t.  Plate. 

Dril'd  Plate. 

4 
i 

4 

! 

Drilled 

O.94O 
0.940 
0.912    —    0.876 
0.892    —    0.871 
O.926 

0-934 
0.998    —    O.SgO 

o  945  -  0.875 

38.12 
38.22 
35-04 

34-44 
35-39 
34-9° 
33-  91 

34-38 

.105 

.108 

.000 

.025 
.126 

.  Ill 

.073 

.096 

0.918 
0.902 

0.965 
0.978 

Drilled  
i  punch,  H  die. 
£  punch,  fif  die. 
Drilled  

Drilled 

8  punch,  I  die.  . 
i  punch,  ||  die. 

Each  result  is  a  mean  of  four,  from  plate  specimens  2,  4,  6  and  8  inches  wide. 
The  pitch  of  rivet  holes  across  the  middle  of  specimens  was  2  inches, 
and  the  width  of  each  specimen  was  so  chosen  that  each  side  passed 
through  the  centre  of  a  rivet  hole,  as  shown  in  Fig.  2.  A  "  ton  "  is  2,240 
pounds.  The  two  diameters  of  punched  holes  are  for  the  two  sides  of 
the  plate. 

A  duplicate  set  of  experiments  on  32  specimens  of  a  some- 
what softer  steel  boiler  plate,  gave  essentially  the  same  results 
(see  "  Engineering,"  6th  May,  1881). 

By  experimenting  on  mild  Fagersta  steel  plates  with  the 
thicknesses  J4,  ^,  ^,  ]/2  and  $£  inch,  Mr.  Kirkaldy  found  the 
ratio  of  the  resistance  of  drilled  specimens  over  that  of  punched 
ones  to  vary  from  about  I.I  (for  *^j,  ^  and  J/6-inch  specimens) 
to  1.5  (for  y%  specimen)  when  unannealed,  and  to  be  about  i.i 
for  all  the  thicknesses  when  annealed.  All  the  specimens  were 
12.5  inches  wide,  with  three  rows  of  0.77  inch  holes,  pitched 
2.5  inches  apart,  running  across  the  specimens.  The  average 
resistance  for  square  inch  of  net  section  was  greater  than  that 


Art.  34.]  PUNCHING,   DRILLING,   ETC.  331 

of  the  original  plate  for  the  drilled  holes,  but  considerably  less 
for  the  punched  ones. 

Mr.  Kirkaldy  states,  "  the  loss  from  punching  is  not  con- 
stant, but  varies  with  the  thickness,  and  also  with  the  hardness 
of  the  material."  He  also  concluded  that  punching  hardens 
the  material  in  the  vicinity  of  the  punch,  and  that  the  effect  of 
punching  is  counteracted  "  to  a  considerable  extent  "  by  an- 
nealing. 

The  results  of  Mr.  Hill's  experiments,  as  given  in  Table 
XXVI.,  show,  for  the  thickness  of  plates  there  used,  that  by 
enlarging  the  diameter  of  the  punched  hole  from  o.i  inch  to 
0.165  inch,  by  reaming,  the  injurious  effect  of  the  punch  is 
entirely  removed. 

Experiments  on  French  steel  plates,  produced  by  the  Bes- 
semer and  Martin  processes  (ntftal fondu),  confirm  this  result 
and*  form  a  basis  for  other  conclusions,  as  follows  ("  The  Use 
of  Steel,"  by  J.  Barba,  A.  L.  Holley,  translator,  p.  40)  : 

"  1st.  That  the  effects  of  punching  and  shearing  are  essen- 
tially local  and  spread  only  over  a  very  restricted  region,  less 
than  0.039  mcn  on  tne  edges  of  the  sheared  or  punched  parts  ; 

"  2d.  That  no  cracks  exist  in  this  altered  region  ; 

"  3d.  That  tempering  destroys  the  effects  of  shearing  and 
punching  by  bringing  the  metal  back  to  the  state  it  would  be 
in  if  drilling  or  planing  had  been  substituted  for  punching  or 
shearing ; 

"  4th.  That  annealing  alone  or  after  tempering  destroys,  as 
tempering  alone  does,  the  alterations  caused  by  shearing  and 
punching." 

These  conclusions  relate  to  plates  from  0.27  inch  to  0.46 
inch  thick. 

In  first-class  practice,  holes  in  steel  plates  and  shapes  are  / 
frequently  first  punched  and  then  reamed  to  a  diameter  0.125  I 
inch  greater. 

Experiments  on  some  narrow  specimens  of  steel  plate  seem 
to  indicate  that  conical  punching  (the  die  0.16  to  0.20  inch 


332  STEEL  IN    TENSION.  [Art.  34. 

greater  in  diameter  than  the  punch)  injures  the  material  less 
than  cylindrical  punching  (with  a  clearance  of  perhaps  T^  inch). 

In  the  working  of  steel  plates  and  shapes,  during  ordinary 
constructive  processes,  all  local  pressure  of  great  intensity,  and 
hammering  while  cold  or  at  a  low  temperature,  tend  to  pro- 
duce internal  strains  of  great  intensity  or  other  changes  in 
molecular  condition  which  cause  the  finished  plate  or  shape  to 
be  liable  to  great  brittleness  and  unlooked-for  failure  of  a  local 
character. 

For  these  reasons  M.  Barba  gives  the  following  directions 
in  regard  to  the  working  of  steel  : 

"  1st.  Avoid  any  local  pressure  of  whatever  nature  it  may 
be ;  2d.  If  local  pressures  have  been  produced  by  blows  of  a 
hammer,  the  action  of  the  punch,  etc.  (which  may,  as  we  have 
seen,  cause  ruptures),  heat  the  piece  to  a  cherry-red  in  a  very 
regular  manner  and  as  much  as  possible  in  its  entirety — the 
whole  of  it  at  once — and  let  it  cool  in  the  open  air  on  a  homo- 
geneous surface,  which  has  all  over  equal  conducting  power. 
This  simple  reheating,  which  may  be  considered  as  annealing 
for  plates  and  bars,  on  account  of  their  slight  thickness,  restores 
to  the  worked  metal  its  original  qualities,  even  if  it  was  in  a 
very  unstable  state  of  equilibrium." 

If  a  large  amount  of  working  (such  as  bending  or  curving) 
of  a  single  kind  is  to  be  done  to  a  single  piece,  it  is  best,  if 
possible,  to  heat  to  a  cherry-red  and  do  the  work  by  stages, 
rather  than  all  at  once  ;  and  then  anneal  after  the  working  is 
completed.  If  the  working  is  local  and  the  heating  irregular, 
it  may  be  necessary  to  anneal  once  or  more  during  the  progress 
of  the  work. 

Local  heating  in  the  production  of  the  ordinary  steel  eye- 
bar  head,  for  example,  frequently  gives  much  trouble,  unless 
resort  is  had  to  subsequent  annealing. 

These  difficulties  in  the  working  of  steel  are  found  more 
pronounced  in  the  higher  grades,  and  much  experience  is  still 
needed  before  they  can  be  entirely  overcome. 


Art.  34.] 


BA  USCHINGER'S  EXPERIMENTS. 


333 


On  account  of  the  homogeneous  character  of  the  metal, 
upsetting  processes,  as  in  riveting,  "etc.,  seem  to  injure  the 
molecular  condition  of  steel  much  less  than  that  of  iron. 

Bauschingers  Experiments  on  the  Change  of  Elastic  Limit  and 
Coefficient  of  Elasticity. 

The  details  of  these  experiments  are  given  in  "  Der  Civil- 
ingenieur,"  Part  5,  1881.  The  manner  of  application  of  the 
tests,  and  remarks  on  the  quantities,  elastic  limit,  stretch  limit, 
and  final  load,  will  be  found  by  referring  to  page  262.  The 
following  is  the  notation  : 

E.  L.  =  elastic  limit  in  pounds  per  square  inch. 
S.-L.  =  stretch  limit  in  pounds  per  square  inch. 

F.  L.  =  final  load  in  pounds  per  square  inch. 

E.     =  coefficient   of  elasticity  in   pounds  per  square 
inch. 

Bessemer  Steel. 


IN    ORIGINAL    CONDI- 

TION. 

AFTER  69  MRS. 

AFTER  0.5  HR. 

AFTER  68  MRS. 

E.  L. 

25,970 

43,272 

8,760 

14,970 

S.-L. 

40,380 

51,920 

55,470 

71,850 

F.  L. 

46,140 

57,690 

70,080 



E. 

29,848,000 

29,549,000 

29,009,000 

30,146,000 

A  small  specimen  of  this  Bessemer  steel,  about  an  inch  in 
diameter,  gave  an  ultimate  resistance  of  75,800  pounds  per 
square  inch. 


334  STEEL  IN   TENSION.  [Art.  34. 

The  elastic  limit  rises  twice  after  two  long  periods  of  rest, 
and  falls  in  a  very  marked  manner  after  the  short  rest  of  0.5 
hour. 

The  stretch  limit  rises  steadily  while  the  coefficient  of  elas- 
ticity falls  twice  and  then  rises  above  its  original  value. 

Prof.  Bauschinger  was  the  first  to  determine,  in  regard  to 
Bessemer  steel,  that  by  stretching  the  metal  beyond  its  elastic 
limit  its  elasticity  is  elevated,  not  only  during  the  time  of 
action  of  the  load,  but  also  during  a  longer  period  of  rest,  with- 
out load,  of  one  or  more  days ;  and  that,  in  this  manner,  the 
elastic  limit  may  exceed  the  load  which  caused  the  stretching. 
(Dingier*  s  Journal,  Band  224.) 

Fracture  of  Steel. 

The  character  of  steel  fractures  has  been  incidentally 
noticed,  in  some  cases,  in  the  different  tables. 

If  the  steel  is  low,  or  if  some  of  the  higher  grades  are 
thoroughly  annealed,  the  fracture  is  fine  and  silky,  provided 
the  breakage  is  produced  gradually.  In  other  cases  the  fract- 
ure is  partly  granular  and  partly  silky,  or  wholly  granular. 

In  any  case  a  sudden  breakage  may  produce  a  granular 
fracture. 

Effect  of  Chemical  Composition. 

The  ten  sets  of  results  given  in  Table  XXVIII.  are  taken 
from  a  great  number  of  similar  ones  established  by  the  United 
States  Test  Board;  "  Ex.  Doc.  23,  House  of  Rep.,  46th  Con- 
gress, 2d  Session."  The  physical  phenomena  developed  in  con- 
nection with  a  given  chemical  constitution  may  be  observed  at 
a  glance. 

The  amount  of  final  contraction  of  fractured  section  may 
be  accurately  estimated  by  comparing  the  ultimate  resistances 
of  the  original  and  final  sections. 


Art.  34.] 


CHEMICAL    COMPOSITION. 


335 


•HI  '6s  >ia<i 
•SHT     'xiwn    oiisvna 

O         O         ^3"        fi        to        0s       W         *"f"       O         O 

en      co"       nT       cT      vcT       M*        cT       u-i      ccT      «T 
M        co       u->       m      o        r^mr^M        w 

-svia  jo  iNaiDidjaoD 

u~>        O         ^        O        *O        co         ^        OJ         to        C^ 

IVNIJ    Jo  UN-JO  H3J 

to     c?     i?    °°     °° 

nil 

ULTIMATE  RESIST- 
ANCE IN  POUNDS  J'ER 
SQ.  IN.  OF 

ll 

O         O         Q         O         O         O         C 
O         O         Q                    O         O         C 

5     8     8    8 

O         CO            M            CO 

M            M           C4           CO         CO          tO^          'T          ^         O           M 

co         w         cocoN         Mxn^ttO 

Original 
Section. 

r^       co"       in        co        O"        o"        co       in        cT       of 

PERCENTAGES  OF 

1 

O        in       O         co       O        oo 

O         O        O         O         O         O 

666666 

C        ~        6 

1 

<J 

CO           CO          4)            o            C>          CO 

O        "-•         u        o        Q        2. 

o      o      g      rt      o      a 
6      6     -^     **      6      6 

C        *-        6 

u 

o        r>i       co        co        co       m 

8     8     8     8     8     8 
666666 

I    §    1 

coo 

li 

§O        CO        O         m        co 
vO         co        -^        T       r^- 

666666 

<N             Cl             M^ 

666 

ll 

\O        vO         t>-        O         W         m 
•f       co        r>«        co        M        co 

CX           CO         O          CO            M            C< 

6      6      6      6      M      M 

C?          S           ? 

MOO 

!i 

0 

M       o       co        O        O        co 

M             Q              M             ^            tO            CO 

O         0         O         O         O         O 

666666 

«     8     § 

0         0         C 

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m       \O        w^        ^       O        O 
~f       t^       co        co        C>       O 

M         O        M        M         M         M 

666666 

^-            M            0 
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666 

if 

*^*       Is**       to       tr>       w^       to 

t^        m        & 

M         rr        O 
0         0        M 

660 

o      o      o      o      o      o 
666666 

if 

1  f  ?  r'  §  j 

o      o      o      £      i:      i 

O^           M 
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coo 
o        * 
coo 

336 


COPPER,    TIN,   ETC.,    IN   TENSION.  [Art.  35. 


The  specimens  were  circular  in  section  and  either  0.625  inch 
or  about  0.8  inch  in  diameter,  while  all  possessed  a  length  of  6 
inches. 


Art.  35. — Copper,  Tin  and  Zinc,  and  their  Alloys. — Phosphor  Bronze. 

Coefficients  of  Elasticity. 

Table  I.  gives  the  coefficients  of  elasticity  (E)  of  the  various 
metals  and  their  alloys,  according  to  the  various  authorities. 
These  coefficients  were  determined  by  experiments  in  tension, 
and  E  is  given  in  pounds  per  square  inch. 

TABLE  I. 


METAL. 

AUTHORITY. 

E. 

REMARKS. 

Brass  

Tredgold. 

8.Q1O  OOO 

Cast  metal. 

Tin 

4  608  ooo 

Zinc 

« 

13  680  ooo 

<«       <« 

Gun  Metal    

«« 

Q  873  OOO 

Copper    8  j  Tin,  i. 

Zinc 

Wertheim. 

12  828  OOO 

Ingot 

Zinc 

12  42O  OOO 

Copper 

ii 

17  7O2  OOO 

ii 

« 

14  958  ooo 

Annealed 

Brass 

« 

12  148  OOO 

ZnCu2. 

"  Berlin  Brass.".  . 

Gun  Bronze  
Alloy 

Thurston. 

13,192,000 

11,468,000 
13  514  ooo 

Copper,  0.90  ;  Tin,  o.  10  (nearly). 
Copper   o  80  •  Zinc   0.20. 

Alloy  . 

ti 

14  286  ooo 

Copper   o  625  '  Zinc   0.375. 

Tobin's  Alloy  .  .  . 
Copper  .  . 

" 

4,545,000 

Q  OQI  OOO 

Composition,  below  table. 
Cast  metal. 

Tobin's  alloy  is  a  composition  of  copper,  tin,  and  zinc,  in 
the  proportions  (very  nearly)  of  58.2,  2.3,  and  39.5,  respectively. 
The  value  of  E  for  this  metal,  and  those  for  the  two  preceding 
and  one  following  it,  are  calculated  for  small  stresses  and 
strains  given  by  Prof.  Thurston  in  the  "  Trans.  Am.  Soc.  Civ. 
Engrs.,"  for  Sept.,  1881. 


Art.  35.] 


COEFFICIENTS   OF  ELASTICITY. 


337 


TABLE   II. 
Cast  Tin. 


>. 

E. 

A 

E. 

1,950 

1,147,000 

3,200 

96,400 

2,360 

472,OOO 

4,000 

41,540 

2,580 

172,000 

Broke  at  4,200  Ibs. 

TABLE   III. 
Cast  Copper. 


/. 

E. 

/. 

E. 

800 

10,000,000 

12,000 

18,750,000 

2,000 

9,091,000 

13,600 

8,I93,OOO 

4,000 

9,091,000 

16,000 

2,235,000 

8,000 

14,815,000 

22,000 

137,000 

Broke  at  29,200  Ibs. 

The  values  of  E  (stress  over  strain)  for  different  intensities 
of  stress  (pounds  per  square  inch)  for  cast  tin,  cast  copper,  and 
Tobin's  alloy,  are  given  in 'Tables  II.,  III.  and  IV. 

"/  "  is  the  intensity  of  stress  in  pounds  per  square  inch,  at 
which  the  ratio  E  exists. 

Each  of  these  metals  is  seen  to  give  a  very  irregular  elastic 
behavior. 

Tables  II.,  III.  and  IV.  are  computed  from  data  given  by 
Prof.  Thurston  in  the  United  States  Report  (page  425)  and 
"  Trans.  Am.  Soc.  Civ.  Engrs.,"  already  cited. 


22 


338 


COPPER,    TIN,   ETC.,   IN   TENSION.  [Art.  35. 


TABLE   IV. 
Tobin's  Alloy. 


/• 

E, 

>. 

E. 

2,000 

4,545,000 

18,000 

5,455,000 

4,OOO 

4,545,000 

24,000 

5,941,000 

6,000 

4,688,000 

30,000 

6,250,000 

8,000 

4,938.000 

40,000 

6,390,000 

10,000 

5,263,000 

50,000 

4,744,000 

14,000 

5,110,000 

60,000 

3,436,000 

Broke  at  67,600  Ibs. 

Ultimate  Resistance  and  Elastic  Limit. 

Table  V.  is  abstracted  from  the  results  of  the  experiments 
of  Prof.  Thurston  as  given  in  the  "  Report  of  the  U.  S.  Board 
Appointed  to  Test  Iron,  Steel  and  other  Metals,"  and  "  Trans. 
Am.  Soc.  of  Civ.  Engrs.,"  Sept.  1881.  The  composition  of  the 
various  alloys  was  as  given  in  the  table,  which  also  contains 
results  for  pure  copper,  tin  and  zinc.  All  the  specimens  were 
of  cast  metal. 

The  mechanical  properties  of  the  copper-tin-zinc  alloys 
have  been  very  thoroughly  investigated  by  Prof.  Thurston 
("Trans.  Am.  Soc.  of  Civ.  Engrs.,"  Jan.  and  Sept.,  1881).  As 
results  of  his  work  he  has  found  that  the  ultimate  tensile  re- 
sistance, in  pounds  per  square  inch,  of  "  ordinary  bronze,  com- 
posed of  copper  and  tin  . . . . ,  as  cast  in  the  ordinary  course  of 
a  brass  founder's  business,"  may  be  well  represented  by  : 

Tc  =  30,000  +  i  ,ooof  ; 
"  where  t  isthe  percentage  of  tin  and  not  above  15  per  cent." 


Art.  35-] 


ULTIMATE  RESISTANCE. 


339 


TABLE  V. 


PERCENTAGE   OF 

POUNDS  STRESS  PER  SQ.  INCH  AT 

PER  CENT.  FINAL 

Copper. 

Tin. 

Zinc. 

Elastic  Limit. 

Ult.  Resist. 

Stretch. 

Contract'n. 

IOO 

00 

00 

11,620 

19,872 

0.05 

10.  0 

IOO 

00 

oo 

11,000 

12,760 

0.005 

8.0 

IOO 

00 

00    . 

14,400 

27,800 

0.065 

15-0 

90 

10 

00 

I5.740 

26,860 

0.037 

13.5 

80 

20 

OO 



32,980 

0.004 

00.0 

70 

30 

OO 

5,585 

5,585 



oo.o 

62 

38 

00 

688 

688 



00.0 

52 

48 

00 

2,555 

2,555 



00.0 

39 

61 

00 

2,820 

2,820 



oo.o 

29 

7i 

00 



1,648 



oo.o 

21 

79 

oo 



4,337 



00.0 

10 

90 

oo 

3,5oo 

6,450 

0.07 

15.0 

00 

IOO 

CO 

1,670 

3,500 

0.36 

75.0 

Queensl'd 

00 

IOO 

oo 



2,760 



47.0 

Banca. 

00 

IOO 

00 

2,000 

3,500 

0.36 

86.0 

Gun 

Bronze. 

90 

10 

00 

IO,OOO 

31,000 

4.6 



80 

oo 

20 



33,140 

32.4 

40.0 

62.5 

00 

37.5 



48,760 

31.0 

29.5 

58.2 

2.3 

39-5 



67,600 

4.0 

8.0 

IOO 

o.o 

0.0 



29,200 

7-5 

16.0 

90.56 

0.0 

9.42 







81.91 

0.0 

17.99 

lO.OOO 

32,670 

31.4 

43-0 

71.20 

o.o 

28.54 

9,000 

30,510 

29.2 

38.0 

60.94 

0.0 

38.65 

16,470 

41,065 

20.7 

28.0 

58.49 

0.0 

41.10 

27,240 

50,450 

10.  1 

17  o 

49.66 

o.o 

50.14 

16,890 

30,990 

5-0 

ii.  5 

41.30 

0.0 

58.12 

3,727 

3,727 



32.94 

0.0 

66.23 

1,774 

i,774 





20.81 

o.o 

77-63 

9,000 

9,000 

o.j6 

0.0 

10.30 

0.0 

88.88 

14,450 

14,450 

0-39 

o.o 

0.0 

0.0 

IOO.OO 

4,050 

5,400 

0.69 

0.0 

70.0 

8-75 

20.25 

18,000  (?) 

31,600 

0.36 

0.0 

57-50 

21.25 

21.25 

1,300 

1,300 





45-0 

23-75 

31.25 

2,196 

2,196 





66.25 

23-75 

10.00 

3,294 

3,394 





58.22 

2.30 

39  48 

30,000  (?) 

66,500 

3-13 

7-o 

IO.OO 

50.00 

40.00 

5.000  (?) 

9-300 

0.7 

0.0 

60.00 

IO.OO 

30.00 

2i,78o(?) 

21,780 

0.15 

o.o 

65.00 

20.00 

15.00 

- 

3,765 

~~ 

340 


COPPER,    TIN,   ETC.,   IN    TENSION. 


[Art.  35. 


TABLE  V.  —  Continued. 


PERCENTAGE   OF 

POUNDS  STRESS  PER  SQ.  INCH  AT 

PER  CENT.  FINAL 

Copper. 

Tin. 

Zinc. 

Elastic  Limit. 

Ult.  Resist. 

Stretch. 

Contract'n. 

70.00 

10.00 

20.00 

24,ooo(?) 

33,140 

o  31 

75.00 

5-oo 

2O.OO 

I2,ooo  (?) 

34,960 

3-2 

5-4 

80.00 

IO  OO 

IO.OO 

12,000  (?) 

32,830, 

1.6 

4.0 

55  oo 

o.  50 

44-50 

22,000 

68,  goo 

9-4 

25-0 

60.00 

2.50 

37-50 

22.000 

57,400 

4-9 

6.6 

72.50 

7-50 

2.00 

11,000 

32,700 

3-7 

ii  .0 

77.50 

12-5 

10.00 

20,000 

36,000 

0.7 

0.0 

85.00 

12.5 

2.50 

I2,OOO  (?) 

34,500 

1-3 

3-o 

The  values  of  the  elastic  limit  in  the  lower  part  of  the  table  were  not  at  all  well 
defined. 


"  For  brass  (copper  and  zinc)  the  tenacity  may  be  taken 


as  : 


T,  =  30,000  +  500^. 


where  z  is  the  percentage  of  zinc  and  not  above  50  per  cent." 

He  found  that  a  large  portion  of  the  copper-tin-zinc  alloys 
is  worthless  to  the  engineer,  while  the  other,  or  valuable  por- 
tion, may  be  considered  to  possess  a  tenacity,  in  pounds  per 
square  inch,  well  represented  by  combining  the  above  formulae 
as  follows : 

Tft  —  30,000  -f  i,ooo/  +  500^. 


These  formulae  are  not  intended  to  be  exact,  but  to  give 
safe  results  for  ordinary  use  within  the  limits  of  the  circum- 
stances on  which  they  are  based. 

Prof.  Thurston  found  the  "  strongest  of  the  bronzes  "  to  be 
composed  of  : 


Art.  35.]  GUN  METAL.  341 

Copper 55-O 

Tin 0.5 

Zinc 44-5 


This  alloy  possessed  an  ultimate  tensile  resistance  of  68,900 
pounds  per  square  inch  of  original  section,  an  elongation  of  47 
to  5 1  per  cent,  and  a  final  contraction  of  fractured  section  of 
47  to  52  per  cent. 

The  first  and  sixth  alloys  of  copper,  tin  and  zinc,  in  Table 
V.,  are  called  by  Prof.  Thurston  "  Tobin's  alloy."  "This 
alloy,  like  the  maximum  metal,  was  capable  of  being  forged 
or  rolled  at  a  low  red  heat  or  worked  cold.  Rolled  hot,  its 
tenacity  rose  to  79,000  pounds,  and  when  moderately  and  care- 
fully rolled,  to  104,000  pounds.  It  could  be  bent  double  either 
hot  or  cold,  and  was  found  to  make  excellent  bolts  and  nuts." 

As  just  indicated  for  the  particular  case  of  the  Tobin  alloy, 
the  manner  of  treating  and  working  these  alloys  exerts  great 
influence  on  the  tenacity  and  ductility. 

Baudrimont  found  for  a  copper  wire  0.0177  inch  in  diameter, 
an  ultimate  resistance  of  about  45,000  pounds  per  square  inch* 
the  wire  being  unannealed,  while  for  a  diameter  of  0.064  inch, 
Kirkaldy  found  about  63,000  pounds  per  square  inch. 

Prof.  Thurston  states :  "  brass,  containing  copper  62  to  70, 
zinc  38  to  30,  attains  a  strength  in  the  wire  mill  of  90,000 
pounds  per  square  inch,  and  sometimes  of  100,000  pounds." 

All  of  Prof.  Thurston's  specimens  were  what  may  be  called 
"long  "  ones,  /.  e.,  they  were  turned  down  to  a  diameter  of  0.798 
inch  for  a  length  of  five  inches,  giving  an  area  of  cross  section 
of  0.5  square  inch. 

Gun  Metal. 

Major  Wade  ("  Reports  of  Experiments  on  Metals  for  Can- 
non," 1856)  made  many  experiments  on  a  gun  metal  composed 
of  copper  89  and  tin  1 1  (very  nearly),  called  gun  bronze. 


342 


COPPER,    TIN,   ETC.,   IN   TENSION. 


[Art.  35. 


He  found  that  different  methods  of  manipulation  of  the 
molten  metal  and  of  treatment,  as  in  cooling,  affected  to  a  great 
extent  its  resistance. 

TABLE  VI. 

Gun  Bronze. 


ULTIMATE   TENSILE   RESISTANCE,    POUNDS   PER  SQUARE   INCH. 

MINUTES   IN 

LADLE. 

Gun-heads. 

Small  bars. 

O 

Highest  

17,698 

17,825 

17,761 

50,973 

31,132 

*5 

Mean  

29,216 

28,775 

28,995 

52,330 

28,153 

29 

Lowest.   ... 

23,381 

24,064 

23.722 

56,786 

28,082 

Density  varied  from  7 . 978  to  8 . 823. 

Table  VI.  gives  the  average  results  of  a  large  number  of 
experiments  made  by  Major  Wade.  It  shows  the  great  range 
in  the  tenacity  of  the  different  specimens. 

General  Results. 

Table-  VII.  gives  general  results  of  various  European  ex- 
perimenters. T  represents  the  ultimate  tensile  resistance  in 
pounds  per  square  inch. 

Some  of  these  results  are  from  the  experiments  of  early  in- 
vestigators, who  attached  little  importance  to  the  size  and  form 
of  the  test  specimen.  In  all  the  cases  the  results  would  be 
more  valuable  if  the  circumstances  of  testing  were  given. 
Those  belonging  to  the  more  unusual  alloys,  however,  possess 
considerable  general  interest  in  spite  of  the  uncertainty  sur- 
rounding their  experimental  origin.  The  presence  of  a  little 
phosphorus  in  copper  is  seen  to  increase  its  resistance  in  a 
marked  manner. 


Art.  35.] 


VARIOUS  ALLOYS. 


343 


TABLE   VII. 


Copper   wrought       •        •  .         

Anderson. 

ii 

« 

« 
ii 

it 

Mallet. 

< 
< 

« 

Anderson. 
i> 

Rennie. 
Stoney. 
Rennie. 
Anderson. 

Everitt. 
« 

Dufour. 
Anderson. 

ii 

ii 

ii 
ii 

ii 

«i 

33,600 
19,000  to  26,100 
16,900 
38,400 
45,400 
47,900 
50,000 

29,000 
30,700 
33,000 
38,100 

36,100 
34,050 
39,650 
30,500 
3,140 
6,950 
5,6oo 

73,000 
96,300 
4,740 
3,000 
18,000 
28,900 
103,000 
80,600 
91,300 
49-300 
7,100 
53,8co 

43,ioo 
54.300 
69,400 
76,200 

85,100 

60,500 
76,200 

85,100 

Copper  cast     

Copper,  bolts  with  phosphorus  o.oi  

Copper  bolts  with  phosphorus  o  015 

Copper  bolts  with  phosphorus  o  02         

Copper  bolts  with  phosphorus  0.03  

Copper,  bolts  with  phosphorus  o  04 

/-Proportions.-^ 

,  Weights  in  100.  > 
Alloy   copper  84  20    tin  15  71  

Alloy   copper  82  81    tin  17   19 

Alloy   copper  81   10    tin  18  go         

Alloy   copper  78  07    tin  21  03    brasses  

Alloy,  copper  34.92,  tin  65.08,  small  bells  
Alloy,  copper  15.  17,  tin  84.83,  speculum  metal 
Tin                      .              

,  Proportion  .  > 

Aluminium  bronze    greatest  

Tin   cast              

Brass   yellow                      .                  

Brass   tube    copper  62    zinc  38 

Brass   tube   copper  70   zinc  30       

Brass   wire              

Sterro-metal    copper  10   iron  TO    zinc  80  . 

Sterro-metal,  copper  60,  iron  3,  zinc  39,  tin  1.5 
Sterro-metal,  copper  60,  iron  4,  zinc  44,  tin  2.0 

Copper  55,  iron   1.77,   zinc  42.36,    tin  0.83  — 

EXPERIMENTER. 


T. 


344  PHOSPHOR  BRONZE  IN   TENSION.  [Art.  35. 


Phosphor  Bronze,  and  Brass  and  Copper  Wire. 

Table  VIII.  contains  the  results  of  the  experiments  of  Mr. 
Kirkaldy  on  phosphor  bronze,  with  two  results  each  for  brass 
and  copper  wire. 

TABLE  VIII. 

Phosphor  Bronze. 


METAL. 

E.L. 

ULT.  RESIST. 
SQUARI 

POUNDS  PER 
E    INCH. 

FINAL  STRETCH. 

Unannealed. 

Annealed. 

Phosphor  bronze  

eer  SOO 

7  5  ,  ooo 

^T,2OO 

74,OOO 

j____ 

4O  ^OO 

6^  700 

26  3OO 

54,100 

21   ^OO 

50,  100 

w  re      

102,750 

49,400 

VI     «? 

« 

121,000 

47,800 

^4.   I 

• 

121  OOO 

e-3  ,400 

42    A. 

i 

I<JQ    TOO 

CA   2OO 

A  A     Q 

« 

mo  ^oo 

c8  QOO 

46  6 

< 

151,100 

64  600 

42.8 

Copper  wire 

63  100 

•37  OOO 

0/1       T 

Brass  wire       .  .              ...        . 

8  1  200 

e  I   CQO 

•^6  ^ 

The  diameter  of  the  phosphor  bronze  wire  varied  from 
about  0.06  inch  to  o.i  I  inch  ;  that  of  the  copper  wire  was  0,064 
inch,  and  that  of  the  brass  wire  0.0605  inch. 

The  final  stretch  is  the  per  cent,  of  the  original  length,  and 
belongs  to  the  annealed  wire. 

The  contraction  of  fractured  section  for  the  phosphor 
bronze  specimens  varied  from  about  four  to  thirty-two  per  cent, 
of  original  area. 


Art.  35.]  EFFECT  OF  HIGH   TEMPERATURE.  345 

The  first  five  results  belong  to  metal  of  the  same  composi- 
tion but  subjected  to  different  treatment. 

Some  specimens  tested  by  Mr.  Kirkaldy  gave  as  low  as 
about  21,700  pounds  per  square  inch. 


Experiments  on  Rolled  Copper  by  the  "  Franklin  Institute 

Committee" 

The  results  of  these  experiments  are  contained  in  the 
"Journal  of  the  Franklin  Institute,"  for  1837. 

That  committee  found,  as  a  mean  of  66  experiments,  the 
ultimate  resistance  of  rolled  copper  to  be  32,826  pounds  per 
square  inch.  The  temperature  of  the  copper  varied  from  62° 
to  82°  FaJir.  "  The  irregularities  of  strength  in  the  different 
specimens  varied  from  1.9  to  4.8  per  cent,  of  the  mean  te- 
nacity." 

The  resistance  was  found  to  be  the  greatest  at  ordinary 
temperatures,  and  to  decrease  with  acceleration  as  the  tempera- 
ture increased. 


Variation   of   Ultimate  Resistance  and  Stretch  at  High 
Temperatures. 

The  results  contained  in  Table  IX.  were  obtained  at  Ports- 
mouth (England)  Dockyard,  and  were  published  in  the  Engi- 
neer, 5th  Oct.,  1877.  "  R"  is  the  ultimate  tensile  resistance  in 
pounds  per  square  inch,  and  "  St."  is  the  per  cent,  of  stretch 
for  a  length  of  10  inches  in  all  except  the  last  (steel)  specimen. 

At  250°  to  350°  the  gun-metal  specimens  lose  about  half 
their  ultimate  resistance  and  nearly  all  their  ductility.  Phos- 
phor bronze  lo£es  about  one-third  of  its  resistance  and  two- 
thirds  of  its  ductility  at  300°  to  400°.  Muntz  metal  and  copper 
are  not  much  affected,  nor  is  cast  iron.  Wrought  iron  and 
steel  gain  in  ultimate  resistance  but  lose  in  ductility.  These 


346 


ALLOYS  IN   TENSION.  [Art.  35. 


GUN  METAL  RODS  I  INCH  IN  DIAMETER. 

in  O  >o 

* 

0                                    0           0           «>. 

8M             VO               0' 
w 

H 

5-                 vS      5-     8 
1                 €     1    £ 

888 

ro  N    IO 
00              M 

^ 

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•" 

OOOOOOOOOOOO 

a    ft   -«    8-   «    sr    fr    !r    -    - 

888 

* 

w        oo         o\        c>       vo'        co'         t^.        el         el         ci 

< 

JOOOOOOOQO 
0          N         t^vSvoco         ^-Q         S 
N           O^M"^OvvOvo           O           t-x 

888 

4 

0>oOOOO                      0                      o 

VQiOTj-CTtf^lO                        O                        O 

< 

OQQOOO                        O                        0 
OQQ-^WVO                        •*                       0 
vo^v5v5^-*-roo^                     vo                      vo^ 

IT)  IO 

t^  r^  10 

S3 

CO                                   CO           O           «»•         O 
CO                                         CO            »0           O"            O* 

* 

JO           0           0           0 
•^*-           •<*•          CO           CO 

vo                                     ~^r          •<£        CO         OO 

Is  I 

31 

lOOOOOOfOCOOO 

NgMOOOCOOOO 

* 

<*        oT         ro        M"         N         fT       co"        >n       vo        vo 

•HHVJ  'aanxvaadwai 

«         o           -cocoooo 

Art.  35.]  EFFECT  OF  HIGH   TEMPERATURE. 


347 


LANDORE  STEEL. 

c 

0 
X 

•   3 
6 

35 

q                           10                 ro                 m                o 

«                                          «                           M                           0                           O 

" 

o                         o                 o                 o                o 
»o                                  S.                6 

3                        4               £               £                ^ 

WROUGHT  IRON. 

Yorkshire. 
Dia.  =  0.70  inch. 

* 

O          fO                     ro                    «o                     O                     CO 

s1    j        E-        ^        ^        JT 

« 

o"       o"                o"                cT                o"                M" 

Remanufac- 
tured 
Dia.  =  o  74  inch. 

* 

OooOOOmiOOOO 

M 

QOOOOOOOOO 
0          >*          M          MOO          moioovo^O 
ioOOO-OMMinir» 

lOVOvo^OVO           t^t>vO           t>fv 

CAST  IRON. 

10  "1  O 
•    N    N    in 

.1  ':  : 

4 

0000000000 

" 

ooooogooQO 
S         r>.Nv5cov5covo         0         Q 
r>.       q.       N        M,       o.       ro       q.      vo       vS.      Q 
o^jj.oo'co       ^f^^^S1^ 

COPPER. 

M 

0 

II 

ej 

Q 

* 

in      in      q       o       q       o       o       o       q       o 

- 

I  1!  1  1  1  Itf  1 

MUNTZ  METAL. 

.s  •  • 

o  J   ; 

8! 

inmo^ioxo        mrofooo        O 

^ 

M"           cS           •*           0           CO"         vo'          vo'            O'           «'           N 
OO            t^COOO            t^l^txt^»>.t^ 

PHOSPHOR  BRONZE. 

3  T 

« 

*?       °.       °.       °.        o        q       q       o        q       q 

< 

|  |  |  |  §  I  |  |  |  | 

•MHVJ  'aanxvaadwax 

i 

348  COPPER,    TIN,  ETC.,   IN   TENSION.  [Art.  35. 

results  would  probably  be  somewhat  varied  by  different  pro- 
cesses of,  and  treatment  in,  manufacture  and  construction. 
The  Muntz  metal  and  copper  specimens  were  rolled. 


Bauschinger*  s  Experiments  with  Capper  and  Red  Brass. 

Prof.  Bauschinger  extended  his  experiments  on  the  repeated 
application  of  stress  so  as  to  cover  not  only  wrought  iron  and 
steel,  the  results  of  which  have  already  been  given,  but  also 
copper  and  red  brass. 

The  notation  is  that  already  used  : 

£.  L.  =  elastic  limit  in  pounds  per  square  inch. 
S.-L.  =  stretch  limit  in  pounds  per  square  inch. 
F.  L.  =  final  load  in  pounds  per  square  inch. 
E.     =  .coefficient  of  elasticity  in  pounds  per  sq.  inch. 
=  "  immediately." 


The  copper  specimens  were  of  rolled  material  about  16 
inches  long  with  a  cross  section  about  2.4  inches  by  0.64  inch. 
These  specimens  gave  an  ultimate  tensile  resistance,  per  square 
inch,  of  28,900  to  32,000  pounds  and  a  final  contraction  of  27 
to  46  per  cent. 

The  red  brass  specimens  were  turned  to  about  one  inch  in 
diameter  and  16  inches  long.  They  gave  ultimate  tensile  re- 
sistances, in  pounds  per  square  inch,  varying  from  19,600  to 
23,460. 

With  one  exception,  in  the  second  case  of  red  brass,  the 
elastic  limit  and  stretch  limit  were  elevated  by  repeated  ap- 
plication of  stress,  whether  immediately  or  at  the  end  of  fol- 
lowing periods  of  rest. 

The  effects  on  the  coefficient  of  elasticity  are  seen  to  be 
somewhat  irregular. 


UNIVERSITY 


Art.  35.]  BAUSCHINGER'S  EXPERIMENTS 

Copper. 


IN    ORIGINAL    CONDI- 

IM'Y. 

IM'Y. 

IM'Y. 

TION. 

E.  L 

5,475 

8,030 

8,790 

H,450 

S.-L. 



11,670 

14,650 

22,880 

F.  L. 

— 

14,600 

21,970 



E. 

16,651,000 

17,249,000 

16,154,000 

15,770,000 

Copper. 


IN    ORIGINAL    CONDI- 
TION. 

AFTER  l8  HRS. 

AFTER  23  HRS. 

AFTER  24  HRS. 

E.  L. 

2,560 

7,320 

8,080 

11,520 

S.-L. 





14,680 

23,040 

F.  L. 



14,650 

22,010 



E. 

l6,OII,000 

16,295,000 

15,197,000 

15,756,000 

Copper. 


IN    ORIGINAL    CONDI- 

TION. 

AFTER  43  HRS. 

AFTER  44.5  HRS. 

AFTER  51.5  HRS. 

E.  L. 

5,840 

8,030 

10,340 

15,390 

S.-L. 



11,670 

14,760 

23,080 

F.  L. 

-^ 

I4,6OO 

22,l6O 



E. 

16,097,000 

l6,7SO,OOO 

l6,O69,OOO 

15,472,000 

350 


COPPER,    TIN,   ETC.,   IN   TENSION.  [Art.  35, 


Red  Brass. 


IN    ORIGINAL    CONDI- 

IM'Y. 

IM'Y. 

TION. 

E.  L. 

7,680 

9,090 

9,260 



S.-L. 

13,960 

16,070 

19,240 



F.  L. 

16,770 

19,550 



E. 

I2,O3O,OOO 

12,485,000 

12,727,000 



Red  Brass. 


IN    ORIGINAL    CONDITION. 

AFTER  17.5  HRS. 

AFTER   21  HRS. 

E.  L. 

5,600 

9,IT5 

8,550 

S.-L. 

14,020 

16,130 

19,240 

F.  L. 

16,820 

19,640 



E. 

12,322,000 

I2,3I4,OOO 

I2,485,OOO 

Red  Brass. 


IN    ORIGINAL    CONDITION. 

AFTER   53  HRS. 

E.  L. 

3,4SO 

9,090 



S.-L. 

13,910 

16,070 



F.  L. 

16,690 

. 



E. 

13,239,000 

12,940,000 



Art.  36.] 


COEFFICIENTS  OF  ELASTICITY. 


351 


The  explanation  of  the  method  of  applying  these  repeated 
stresses  will  be  found  in  connection  with  the  results  for 
wrought  iron  on  page  262. 

Art.  36. — Various  Metals  and  Glass. 
Coefficients  of  Elasticity. 

The  following  values  of  the  coefficients  of  elasticity,  in 
pounds  per  square  inch,  contained  in  Table  I.  are  taken  from 
Wertheim's  "  Physique  Mtcanique"  pages  57  and  58.  The  CO- 


TABLE  I. 


COEFFICIENT   C 

F   ELASTICITY. 

Drawn. 

Annealed. 

Lead  

Wertheim 

2  564  OOO 

2  4.^7  OOO 

Cadmium 

« 

7  **n  OOO 

7  ^^^  OOO 

Gold  

« 

ii  «;64,ooo 

7  Q42  OOO 

Silver 

14 

10  463  ooo 

10  155  ooo 

Palladium  

II 

16  721  ooo 

I3,g2O,OOO 

Platinum  

M 

24.  2^7  OOO 

22  067  OOO 

efficients  are  the  means  of  a  large  number  of  tensile  experi- 
ments, with  the  exception  of  that  for  cadmium,  which  was 
derived  from  experiments  on  transverse  vibrations.  This  last 
method  gave  results  which  differed,  in  most  cases,  from  the 
direct  tensile  ones  not  more  than  the  latter  did  from  each 
other. 

Wertheim  also  gives  for  the  tensile  coefficients  of  elasticity 
of  some  different  glasses : 


352  METALS  AND   GLASS  IN   TENSION.  [Art.  36. 

Mirror  glass E  =  8,792,000  pounds  per  square  inch. 

Goblet  (common) .£  =  9,559,000       "         "        "         " 

Goblet  (fine) .£•  =  8,589,000       "         "        "         " 

Goblet  (violet) .£  =  7,110,000       "         "        "         " 

"Crystal" E  =  5,830,000       " 


Ultimate  Resistance  and  Elastic  Limit. 

Wertheim  determined  the  elastic  limit  of  many  of  the 
more  rare  metals,  such  as  those  named  in  Table  L,  and  they 
are  here  given  in  pounds  per  square  inch  : 

ANNEALED.  DRAWN. 

Lead 284  to  355 

Cadmium 142  to  171 

Gold 4,266  to  19,200 

Silver 4,266  to  16,350 

Palladium 7, no  to  25,600 

Platinum 20,600  to  37,000 

His  "  limit  of  elasticity  "  is  that  force  which  will  perma- 
nently elongate  the  metal  0.000,05  of  its  original  length,  and 
all  his  experiments  were  made  on  wires  of  very  small  di- 
ameters. 

The  following  ultimate  resistances  were  found  for  wires 
about  -jVth  inch  in  diameter  by  Baudrimont  ("  Annales  de 
Chimie,"  1850): 

Gold 17,100  to  26,200  pounds  per  square  inch. 

Silver 40,3001040,550       "         "        "          " 

Platinum 32,3001032,700       "         "        "          " 

Palladium 51,7501052,640       "         "        "          " 

The  ultimate  resistances  of  some  other  metals  are : 

METAL.  EXPERIMENTER.  ULT.  RESIST. 

Cast  lead Rennie 1,824  pounds  per  square  inch. 

Sheet  lead Navier   1,926       "         "        "         " 

Pipe  lead Jardine 2,240       "         "        "         " 

Soft  solder  (|  tin,  \  lead).  Rankine 7,500       " 


Art.  37.]  CEMENT  AND  BRICK.  353 

Sir  Wm.  Fairbairn  ("  Useful  Information  for  Engineers," 
second  series,  pages  226  and  267)  found  the  following  ultimate 
resistances  in  pounds  per  square  inch  by  direct  pull  on  straight 
tensile  specimens  : 

Flint  glass 2,413  pounds. 

Green  glass 2, 896       ' ' 

Crown  glass 2,546       " 

The  specimens  were  of  circular  section  and  about  0.53  inch 
in  diameter. 

By  subjecting  spherical  glass  shells  to  internal  pressure  he 
found  the  following  ultimate  resistances  in  pounds  per  square 
inch  : 

Flint  glass 4,200  pounds. 

Green  glass 4, 800       ' ' 

Crown  glass 6,000       " 

The  thickness  of  these  shells  varied  from  about  0.02  (crown 
and  green  glass)  to  0.08  (flint  glass)  inch. 

Art.  37. — Cement,  Cement  Mortars,  etc. — Brick. 

The  ultimate  tensile  resistance  of  these  materials  depends 
upon  many  circumstances,  and  only  a  few  out  of  a  great  num- 
ber of  experimental  results  will  be  given.  These  results  will 
be  so  chosen  as  to  be  representative,  but  a  full  arid  detailed 
knowledge  of  the  action  of  cements  and  cement  mortars,  under 
different  circumstances  of  testing  and  variety  of  composition, 
must  be  acquired  by  an  examination  of  the  original  memoirs. 

Mr.  Bremermann,  during  the  construction  of  the  St.  Louis 
bridge,  found  in  18  experiments  with  pure  "  Fall  City"  (Louis- 
ville) cement :  Elastic  limit,  16  to  104  pounds  per  square  inch, 
with  a  mean  of  72  ;  ultimate  resistance,  35  to  147  pounds  per 
square  inch,  with  a  mean  of  no;  coefficient  of  elasticity, 
800,000  to  6,930,000  pounds  per  square  inch,  with  a  mean  of 

2,239,000. 

23 


354 


CEMENT  AND  BRICK  IN   TENSION.  [Art.  37. 


TABLE   I. 


KIND   OF   CEMENT. 

NO. 
TESTS. 

C. 

NO. 
TESTS. 

T 

Toepffer  Grawitz  &  Co    Stettin   Germany 

216 

Hollick  &  Co    London                                                           

216 

Wouldham  Cement  Co    London 

Saylor's  Portland  Cement,  Coplay,  Pennsylvania  
Wampun  Cement  &  Lime  Co.,  Newcastle,  Pennsylvania  

8 

12 

1,078 
968 

2 

3 

184 

1  68 

1^8 

1 

6 

93i 

U 

Francis  &  Co                  

i 

163 

L-n 

Wm    McKay  Ottawa  Canada 

882 

G 

Borst  &  Roggenkamp,  Delfzyl,  Netherlands  
Louque'ty  &  Co     Boulogne-sur-mer,  France. 

12 

826 
764. 

3 

132 

i  8 

t 

Riga  Cement  Co.,  Riga,  Russia  

f. 

693 

— 

Scanian  Cement  Co    Lomma   Sweden                 .... 

606 

Bruno  Hofmark  Port-Kund.  Esthland,  Russia 

*6 

583 

Coplay  Hydraulic  Cement,  Coplay  Cement  Co.,  Coplay,  Pa  .  . 
Charles  Tremain  Manlius  N.  Y  .     . 

8 

292 
276 

2 

w 

38 

j 

Allen  Cement  Co.,  Siegfried's  Bridge,  Pennsylvania  

276 

a 

P  .  Gouvreau  Quebec    Canada 

8 

1 

Riga  Cement  Co.  ,  Riga,  Russia  

6 

44 

GJ 

Anchor  Cement  Co.,  Coplay,  Pa 

208 

Cumberland  Hydraulic  Cement  Co.,  Cumberland.,  Md  
Societd  Anonyme   France  .... 

12 

I56 
184 

3 

41 

| 

Howe's  Cave  Cement,  No.  i,  Howe's  Cave,  N.  Y  

41           u           "         No  2               " 

6 

s 

| 

28 

1 

No.  3,               "                      
Societal  Anonima  Emilia  Italy  ist  quality 

8 

170 

181 

2 

31 

i 
c 

a 

'•           **                 4k          "        2d  quality                     

B. 

Thomas  Gourdy,  Limehouse,  Ontario,  Canada  

8 

126 

3 

23 

c 

A.  H.  Lavers,  London     

6 

I 

* 

Scott's  Selenitic  Cement  (Howe's  Cave  lime  and  plaster)  
Parian  Cement,  Francis  &  Co.,  London,  ist  quality  

20 
8 

2C,8 

I  175 

5 

52 
181 

"           "               "           "             u         2d  quality  . 

606 

1  60 

u         A.  H.  Lavers,         "        

6 

Table  I.  contains  the  results  of  the  tests  made  by  General 
Q.  A.  Gillmore  at  the  Centennial  Exposition,  Philadelphia,  in 
1876  (Van  Nostrand's  Magazine,  March,  1877).  The  speci- 
mens were  prepared  "  by  mixing  them  dry  in  each  case  with 
an  equal  measure  of  clean  sand,  tempering  the  mixture  with 
water  to  the  consistency  of  stiff  mason's  mortar,  and  then 
moulding  it  into  briquettes  of  suitable  form  for  obtaining  the 
tensile  strength  in  a  sectional  area  i^  inches  square,  equal  to 
2^  square  inches.  The  briquettes  were  left  in  the  air  one 
day  to  set,  then  immersed  in  water  for  six  days,  and  tested 
when  seven  days  old.  After  thus  obtaining  the  tensile  strength 


Art.  37.] 


PHILBRICK'S  EXPERIMENTS. 


355 


in  each  case,  the  ends  of  the  broken  specimens  were  ground 
down  to  \l/2  inch  cubes,  which  were  used  the  same  day  for 
obtaining  the  compressive  strength  by  crushing."  The  col- 
umns *'  No.  tests  "  give  the  number  of  experiments  from  which 
were  obtained  the  mean  values  contained  in  the  columns  C 
and  T. 

C  —  Ult.  Compressive  resistance  in  Ibs.  per  sq.  in. 
T=     "     Tensile  "  "    "       "     "     " 

The  former  is  given  here  in  order  to  avoid  the  repetition  of 
the  makers'  names  hereafter. 

TABLE    II. 


AGE. 

N.  R. 

7*. 

N.  N.  J. 
T. 

N.  N.  J. 

T. 

N.  Y.  R. 

T. 

L.  H. 
T. 

H.  C. 

T. 

16  hours  

47 

24  hours   .... 

c  e 

6l 

<;8 

d$ 

44 

60 

36  hours  

c  c 

48  hours  

60 

64 

7Q 

CQ 

4Q 

60  hours  
72  hours   .... 

68 



77 

4  days  .  . 

7-1 

7  days  

97 



7i 

85 
08 

joC 



14  days  .  . 

1  20 

v° 

CQ 

19  days  
21  days  

140 

*34 



Q7 



25  days 

117 

I  month  
64  days  
69  days 





92 

297 

^2C 

108 

2  months  
3  months  
3i  months.  .  .  . 
4  months  
6  months  
i  year  





157 

202 
221 

250 
38l 

175 
177 

1  1  year 

241 



2  years  

2?7 

385 

336 

266 

292 

All  specimens  of  neat  cement  mixed  with  fresh  water  at  about  60°  Fakr. 


356 


CEMENT  AND  BRICK  IN   TENSION.  [Art.  37. 


Table  II.  contains  the  results  of  a  large  number  of  tests  of 
Rosendale  cement  (Trans.  Am.  Soc.  of  Civ.  Engrs.,  Vol.  VII., 
Feb.  1878 — "  Improvement  of  the  South  Boston  Flats,"  by 
Edward  S.  Philbrick),  also  that  from  Howe's  Cave. 

"  N.  R."  signifies  "  Newark  &  Rosendale  Cement  Co." 
"  N.  N.  J."  signifies  "  Newark,  N.  J.,  Lime  &  Cement  Co." 
"  N.  Y.  R."    signifies    "  New   York  &  Rosendale  Lime    & 
Cement  Co." 

"  L.  H."  signifies  "  Lawrence  Cement  Co. — Hoffman  Rosen- 
dale  Cement." 

"  H.  C."  signifies  "  Howe's  Cave  Cement." 

T  =  ultimate  tensile  resistance  in  pounds  per  square  inch. 

The  values  of  T  are  averages  of  from  I  to  2,217  tests. 
Table    III.  gives   some   results    of   the  same  neat  cement 
specimens  mixed  with  salt  water  at  about  60°  Fahr. 


TABLE   III. 


N.    N.    J. 

L.    H. 

N.    N.  J. 

L.    H. 

AGE. 

AGE. 

T. 

T. 

T. 

T. 

24  hours.  .  .  . 

39 

•9 

2    months.  . 



134 

2  days  



45 

34  months.  . 



275 

7  days  .  . 

56 

6    months 

2^7 

10  days 

CQ 

oft? 

20  days  



63 

1  1  year  .... 

205 



I  month  .  .  . 

105 

79 

2    years.  .  .  . 

3" 

234 

The  notation  is  the  same  as  that  of  Table  II. 


Art.  37.] 


MACLA  Y'S  EXPERIMENTS. 


357 


Experiments  and  Conclusions  of  Wm.  W.  Maclay,  C.  E. 

The  following  tables  and  conclusions  are  abstracted  from 
"  Notes  and  Ekperiments  on  the  Use  and  Testing  of  Portland 
Cement,"  by  Wm.  W.  Maclay,  C.  E.  (Trans.  Am.  Soc.  of  Civ. 
Engrs.,  Vol.  VI.,  Dec.,  1877). 

He  made  many  valuable  experiments  in  order  to  determine 
the  effect  of  different  temperatures  at  different  periods  in  the 
life  of  the  cement. 

TABLE   IV. 
Portland  Cement. 


AVERAGE  TENSILE  STRENGTH  PER  SQ.  IN. 


TEMP.  (FAHR.)  OF 
IN  WHICH  BRIQ 
WERE  IMMERSED 

7  days  old. 

Temperature  of  cement  paste  when  bri- 
quettes were  moulded. 

21  days  old. 

Temperature  of  cement  paste  when  bri- 
quettes were  moulded. 

32° 

40° 

50° 

60° 

70° 

32° 

40° 

50° 

60° 

70° 

40° 
50° 
60° 

70° 

Lbs. 
156 
186 

259 
299 

Lbs. 
147 
206 
275 
3H 

Lbs. 
131 

183 
245 
299 

Lbs. 
133 
194 
240 
286 

Lbs. 
143 

I9T 

254 

Lbs. 
360 

Lbs. 
307 

Lbs. 
236 
292 
3I8 
403 

Lbs. 
244 
260 

309 
386 

Lbs. 

212 

251 
282 
336 

40° 

40° 

45° 

45° 

45° 

46° 

46° 

43° 

43° 

43° 

Temperature  of  air  during  last  24  hours. 

Table  IV.  contains  some  of  the  results  of  Mr.  Maclay's 
experiments  which  were  made  to  determine  the  effect  of  the 
temperature  of  the  water  in  which  the  specimens  of  neat 
cement  were  mixed.  After  the  briquettes  were  moulded  at 
temperatures  shown  in  the  upper  horizontal  column,  and  im- 
mersed in  water  either  7  or  21  days  at  the  temperature  shown 


CEMENT  AND  BRICK  IN   TENSION.  [Art.  37. 

in  the  left  vertical  column,  they  were  taken  out  and  dried  in 
air  at  the  temperature  shown  in  the  lower  horizontal  row. 
The  resistances  in  pounds  are  averages  of  five  or  more  results. 

From  these  and  many  other  similar  results,  Mr.  Maclay 
concluded  that  the  ultimate  resistance  follows  "  very  closely 
the  temperature  of  the  water  in  which  the  sample  briquettes 
were  kept  immersed,  the  warmest  water  giving  the  greatest 
tensile  strength  ;  that  a  change  from  40°  to  70°  in  the  water, 
increases  the  tensile  strength  of  the  briquettes  of  neat  cement, 
seven  days  old,  from  63  to  168  pounds  per  square  inch,  or  from 
33  to  127  per  cent. ;  of  the  briquettes  of  mortar,  gauged  I  to  I, 
the  same  change  in  temperature  increases  them  from  32  to  59 
pounds  per  square  inch,  or  from  87  to  133  per  cent.  ;  of  the 
briquettes  gauged  I  cement  to  2  of  sand,  from  19  to  37  pounds 
per  square  inch,  or  from  95  to  176  per  cent.  After  an  interval 
of  three  weeks  the  changes  in  tensile  strength,  .  .  .  become 
less  marked,  and,  in  some  cases,  an  increase  in  the  temperature 
of  the  water  diminishes  the  tensile  strength." 

Other  experiments  seemed  to  "  show  quite  conclusively 
that  the  tensile  strength  increases  directly  as  the  temperature 
of  the  air  when  the  cement  is  being  gauged,  and  inversely  as 
the  temperature  of  the  air  to  which  it  is  exposed  for  the  last 
24  hours  before  breaking." 

"  Exposing  the  briquettes  after  six  days'  immersion  in 
water  to  a  high  drying  temperature  weakens  them  so  invariably, 
that  some  interference  with  the  setting  seems  clearly  demon- 
strated  " 

Some  further  experiments  led  him  to  conclude  "  that  Port- 
land cement  gauged  with  either  fresh  or  salt  water,  hardens 
more-  rapidly  when  immersed  in  salt  or  sev/er  water  than  in 
fresh  water  for  the  first  seven  days,  and  that  this  increase  in 
the  tensile  strength  probably  continues  for  at  least  a  year, 
more  rapidly  in  the  one  than  in  the  other.'' 

Table  V.  contains  the  results  of  Mr.  Maclay' s  tests  on  the 
relative  influence  of  fine  and  coarse  sand. 


Art.  37.] 


MA  CLAY'S  EXPERIMENTS. 


359 


TABLE    V. 

Portland  Cement. 

Showing  difference  in  mortars  with  fine  and  coarse  sand. 


I  VOL.    CEMENT. 

I  VOL.  CEMENT. 

I  VOL.   CEMENT. 

AGE   OF 

I  VOL.    SAND. 

2  VOLS.   SAND. 

3  VOLS.   SAND. 

Fine  sand. 

Coarse  sand. 

Fine  sand. 

Coarse  sand. 

Fine  sand. 

Coarse  sand. 

i  week.  .  .  . 

85 

95 

33 

63 

19 

41 

i  month.  .  . 

162 

202 

Si 

94 

49 

74 

TABLE   VI. 

Portland  Cement. 


w. 

BRAND. 

T. 

P.  c. 

101.5 

Alsen  &  Son,  Itzehoe,  Germany. 

326 

93 

108.0 

«             «           «               <* 

340 

91 

112   0 

Burham. 

?89 

87 

II3.0 

1C 

317 

87 

II4.O 

« 

285 

88 

II5.0 

Gibbs. 

280 

90 

116.0 

Burham. 

316 

88 

117.0 

" 

301 

85 

118.0 

« 

276 

85 

119.0 

ii 

305 

85       ' 

120.0 

" 

252 

84 

121  .O 

Saylor's  American  Portland. 

269 

90 

122.0 

<(               «               « 

281 

90 

123.0 

«<               «               «< 

272 

90 

I24.O 

«               «<               « 

260 

90 

126.0 

«               «i               <  > 

265 

90 

128.0 

n               <t               « 

369 

87 

132.0 

322 

78 

W     —  weight  in  Ibs.  per  bushel. 

T       =  ultimate  tensile  resistance  in  Ibs.  per  sq.  in. 

P.  c.  —  percentage  that  passed  through  sieve  of  2,500  meshes  per  sq.  in. 


CEMENT  AND  BRICK  IN   TENSION.  [Art.  37. 


"  The  deduction  from  this  table  is,  that  by  increasing  the 
fineness  of  the  sand  of  which  the  mortar  is  made  the  tensile 
strength  is  diminished,  and  that  this  reduction  in  tensile 
strength  increases  with  the  amount  of  sand  used  in  the  mortar." 
An  English  experimenter,  Lieut.  W.  Innes,  R.  E.,  was  led  to 
the  same  conclusion. 

Table  VI.  shows  the  results  of  experiments  by  Mr.  Maclay, 
made  to  determine  the  connection  between  the  weight  per 
bushel  and  tensile  resistance  of  specimens  seven  days  old. 
Commenting  on  the  results  he  says,  "  The  close  connection  be- 
tween the  weight  .  .  .  and  the  tensile  strength  .  . 
is  now  proved  to  be  very  uncertain,  if  not  entirely  fallacious." 
As  will  hereafter  be  seen,  these  experiments  invalidate  a  con- 
trary conclusion  reached  by  the  English  experimenter,  John 
Grant,  M.  Inst.  C.  E.,  in  1864. 

.  TABLE   VII. 


S 

< 

>< 
W 

GILLING- 

J.   B.  WHITE 

FRANCIS 

« 

FRANCIS. 

O 
£ 

PORTLAND. 

SAYLOR'S 

D 

« 

P 

HAM. 

BROS. 

BROS. 

AGE. 

Port- 
land. 

Port- 
land. 

Port- 
land. 

Port- 
land. 

Neat 
cement. 

i  cement 
i  sand. 

Roman. 

Medina. 

American 
Portland. 

i  week. 

?78 

184 

212 

250 

363 

i57 

90 

94 

364 

2      u 

256 

i75 

182 

248 

77 

J35 

3       " 
i  mo  .  . 

359 
332 

302 
374 

306 

30  r 

380 

400 

416 

201 

83 
IXO 

132 
136 

4i3 

2 

5^4 

416 

378 

431 









498 

3 

525 

423 

383 

497 

469 

243 

i43 

2CO 

525 

4 

5*3 

432 

428 

529 







584 

I    : 

554 
355 

459 
327 

426 
264 

SI2 

312 

523 

284 

210 

l8j 

576 

575 

I    : 





, 











586 

9 

3°4 

310 

270 

404 

542 

308 

209 

203 

10 

. 





•  

599 

i    yr. 

443 

414 

367 

569 

54* 

318 

286 

212 

i*       - 

611 

497 

339 

569 







2 

326 

355 

224 

476 

589 

351 

24| 

123 



3 





584 

349 

268 

122 



4 









583 

364 

281 

128 



5 



- 





580 

364 

279 

136 



6 









580 

364 

296 

162 



7 



59° 

384 

315 

1  68 

Vertical  columns  give  tensile  resistance  in  pounds  per  square  inch. 


Art.  37.] 


KEENE  AND  PARIAN  CEMENTS. 


361 


Table  VII.  shows  the  variation  of  ultimate  tensile  resist- 
ance, per  square  inch,  of  various  neat  cements  (with  one  ex- 
ception) with  age.  The  left  vertical  column  shows  the  time 
during  which  the  specimens  were  kept  under  water,  and  the 
other  vertical  columns  the  ultimate  tensile  resistance  in  pounds 
per  square  inch. 

The  results  for  the  Burham,  Francis,  Tingey,  Gillingham 
and  Saylor  Portland  cements  are  from  Mr.  Maclay's  paper ; 
the  others  are  from"  Experiments  on  the  Strength  of  Cements," 
London,  1875,  by  John  Grant,  M.  Inst.  C.  E.  ;  all  are  means 
of  great  numbers  of  experiments. 

The  mean  results  of  Mr.  John  Grant's  experiments  on 
Keene's  and  Parian  cements  are  given  in  Table  VIIL 

TABLE   VIII. 


KEENE'S 

CEMENT. 

PARIAN   ( 

:EMENT. 

AGE   AND   TIME   IM- 
MERSED  IN   WATER. 

In  water. 

Out  of  water. 

In  water. 

Out  of  water. 

T. 

f. 

r. 

T. 

I  week  . 

242 

24^ 

264 

28* 

2       "        

216 

26o 

26? 

2o8 

•j       " 

22J. 

2^8 

242 

OTO 

I  month. 

218 

260 

242 

002 

2      "        

2O2 

288 

222 

022 

•*      " 

226 

•J2O 

2^2 

q8o 

T  =  ultimate  tensile  resistance  in  pounds  per  square  inch. 

As  the  result  of  his  experiments  on  Portland  and  Roman 
cements  Mr.  Grant  was  led  to  the  following  conclusions : 


362  CEMENT  AND  BRICK  IN   TENSION.  [Art.  37. 

1.  Portland  cement,  if  it  be  preserved  from  moisture,  does  not,  like  Roman 
cement,  lose  its  strength  by  being  kept  in  casks,  or  sacks,  but  rather  improves  by 
age  ;  a  great  advantage  in  the  case  of  cement  which  has  to  be  exported. 

2.  The  longer  it  is  in  setting,  the  more  its  strength  increases. 

3.  Cement  mixed  with  an  equal  quantity  of  sand  is  at  the  end  of  a  year  ap- 
proximately three-fourths  of  the  strength  of  neat  cement. 

4.  Mixed  with  two  parts  of  sand,  it  is  half  the  strength  of  neat  cement. 

5.  With  three  parts  of  sand,  the  strength  is  a  third  of  neat  cement. 

6.  With  four  parts  of  sand,  the  strength  is  a  fourth  of  neat  cement. 

7.  With  five  parts  of  sand,  the  strength  is  about  a  sixth  of  neat  cement. 

8.  The  cleaner  and  sharper  the  sand,  the  greater  the  strength. 

9.  Very  strong  Portland  cement  is  heavy,  of  a  blue-gray  color,  and  sets  slowly. 
Quick  setting  cement  has,  generally,  too  large  a  proportion  of  clay  in  its  composi- 
tion, is  brownish  in  color,  and  turns  out  weak,  if  not  useless. 

TO.  The  stiffer  the  cement  is  gauged,  that  is,  the  less  the  amount  of  water  used 
in  working  it  up,  the  better. 

11.  It  is  of  the  greatest  importance  that  the  bricks,  or  stone,  with  which  Port- 
land cement  is  used,  should  be  thoroughly  soaked  with  water.     If  under  water,  in 
a  quiescent  state,  the  cement  will  be  stronger  than  out  of  water. 

12.  Blocks  of   brick-work,  or   concrete,   made  with   Portland   cement,  if   kept 
under  water  till  required  for  use,  would  be  much  stronger  than  if  kept  dry. 

13.  Salt  water  is  as  good  for  mixing  Portland  cement  as  fresh  water. 

14.  Bricks  made  with  neat  Portland  cement  are  as  strong  at  from  six  to  nine 
months  as  the  best  quality  of  Staffordshire  blue  brick,  or  similar  blocks  of  Bramley 
Fall  stone,  or  Yorkshire  landings. 

15.  Bricks  made  of  four   parts  or  five   parts  of  sand  to  one  part  of  Portland 
cement  will  bear  a  pressure  equal  to  the  best  picked  stocks. 

16.  Wherever  concrete  is  used  under  water,  care  must  be  taken  that  the  water 
is  still.     Otherwise,  a  current,  whether  natural  or  caused  by  pumping,  will  carry 
away  the  cement,  and  leave  only  the  clean  ballast. 

17.  Roman  cement,  though  about  two-thirds  the  cost  of  Portland,  is  only  about 
one-third  its  strength,  and  is  therefore  double  the  cost,  measured  by  strength. 

1 8.  Roman  cement  is  very  ill  adapted  for  being  mixed  with  sand. 

Mr.  Don  J.  Whittemore  has  proposed  the  following  formula 
for  the  ultimate  tensile  resistance  of  cements  : 


in  which    T  is  the  ultimate  tensile  resistance  in  pounds   per 
square  inch  ;  A,  an  empirical  coefficient,  and  N  the  age  of  the 


Art.  37.] 


ARTIFICIAL   STONES. 


363 


cement  in  days.     For  Portland  cement  (up  to  two  years  old) 

he  gives  x  —  10,  and  A  = 

267  to  356,   by  the  aid  of 

Mr.     Grant's    experiments. 

(See  Trans.   Amer.  Soc.  of 

Civ.  Engrs.,  Vol.  VII.,  Sept. 

1878). 

Fig.  I  shows  the  bri- 
quette used  by  Mr.  Maclay; 
Fig.  2,  that  used  by  Mr. 
Grant,  while  that  shown  in 
Fig.  3  is  the  one  generally 
used  at  the  present  time. 
Each  briquette  is  \y2  inches 
thick,  giving  a  breaking  sec- 


Fig.1 


Fig.2 


tion  of  \y2  X  Ijfc  =  2.25 
square  inches.  In  such  test- 
ing it  is  very  necessary  that  the  pull  should  be  central. 


Artificial  Stones. 

The  tensile  resistances  of  many  artificial  stones  and  some 
natural  British  ones,  can  be  found  in  "  A  Practical  Treatise  on 
Natural  and  Artificial  Concrete,"  by  Henry  Reid,  London, 
1879. 

On  page  198  he  gives  the  following  results  of  Professor 
Ansted's  experiments,  T  being  the  ultimate  resistance  per 
square  inch  : 

Ransome  stone  (artificial) . T  =  360  pounds. 

Portland  stone T  —  201       " 

Bath  stone T  =  145      " 

Caen  stone T  —  140      " 


He  also  gives  for  "  Victoria  "  (artificial)  stone,  three  months 


old, 


364  CEMENT  AND  BRICK  IN   TENSION.        '    [Art.  37. 

T  —  740  pounds  per  square  inch. 

From  35  experiments  on  "  rock  concrete  "  pipe  two  years 
old,  for  drainage  and  sewage  purposes,  Mr.  Reid  found : 

HIGHEST.  MEAN.  LOWEST. 

T  =  700 444 213  pounds  per  square  inch. 

,  Bricks. 

Mr.  Francis  Collingwood,  C.  E.  (Trans.  Amer.  Soc.  of  Civ. 
Engrs.,  Vol.  VII.,  Sept.,  1878),  found,  as  a  result  of  twelve 
experiments  on  "  good  Haverstraw  stock  brick,"  the  following 
values : 

HIGHEST.  MEAN.  LOWEST. 

T  =  358 169 90  pounds  per  square  inch. 

Adhesion  between  Bricks  and  Cement  Mortar. 

General  Q.  A.  Gillmore  ("  On  Limes,  Hydraulic  Cements 
and  Mortars  ")  cemented  Croton  bricks  together  crosswise  and 
then  separated  them  by  a  pull.  He  used  pure  cement  paste 
and  mortars  of  various  proportions,  by  volume,  of  cement  to 
sand,  but  never  more  sand  than  I  volume  of  cement  to  2  vol- 
umes of  sand.  Nearly  all  the  cement  was  Rosendale,  although 
some  specimens  were  prepared  with  Hancock  (Maryland)  or 
James  River  cement.  Bricks  so  cemented  in  pairs  were  kept 
320  days  and  then  separated.  Reviewing  the  results,  Gen. 
Gillmore  says,  "  In  tearing  the  bricks  apart,  at  the  expiration 
of  the  time  specified,  in  a  majority  of  cases  the  surface  of  con- 
tact of  the  brick  and  mortar  remained  intact,  the  adhesion  to 
the  brick  overcoming  the  cohesive  strength  either  of  the  bricks 
themselves,  or  of  the  mortar  composing  the  joint  between 
them.  The  results,  therefore,  although  interesting  for  other 
reasons,  furnish  no  entirely  satisfactory  measure  of  the  power 
of  adhesion." 


Art.  38.] 


ADHESION   TO  BRICKS. 


365 


Also,  "  At  the  age  of  320  days  (and  perhaps  considerably 
within  that  period)  the  cohesive  strength  of  pure  cement  mor- 
tar exceeds  that  of  Croton  front  bricks.  The  converse  is  true 
when  the  mortar  contains  fifty  per  cent.,  or  more,  of  sand/' 

TABLE   IX. 


RATIO  OF  ADHESION 

ADHESION    PER 

MORTAR    OR    PASTE. 

MATERIALS    CEMENTED. 

TO  RESISTANCE  OP 

SQ.    INCH    IN    LBS. 

PURE    CEMENT. 

Pure  cement  

Croton  bricks. 

30.8 

I.OO 

vol.  cement    I  vol.  sand 

15-7 

0.51 

* 

2 

12-3 

0.40 

4 

3 

6.8 

O.22 

* 

4 

5-2 

0.17 

' 

5 

4-3 

0.14 

1 

6 

3-3 

O.II 

3ur 

^  cem 

nt 

Fine  cut  gr 

nite. 

27    % 

I  .OO 

vc 

1.  cen 

ent    I  v 

1.  sa 

d 

T*      3 

20.8 

0.76 

' 

2 

«        « 

12.6 

0.46 

' 

3 

«        « 

9.2 

o-33 

4 

7-9 

0.29 

Table  IX.  contains  the  results  of  another  series  of  experi- 
ments by  General  Gillmore,  made  for  the  purpose  of  determin- 
ing the  adhesion  to  Croton  front  bricks  and  fine  cut  granite, 
of  mortars  containing  different  proportions  of  sand.  "  The 
bricks  were  used  wet,  and  were  well  pressed  together  by  hand. 
They  were  wetted  with  fresh  water  every  alternate  day  for  29 
days,  the  age  of  the  mortar  when  tested.  Each  result  is  the 
average  of  five  trials." 


Art.  38.— Timber. 

Table  I.  contains  the  results  of  experiments  made  by  Che- 
vandier  and  Wertheim  ("  Memoire  sur  les  Propri£t6s  M£canique 
du  Bois;"  by  E.  Chevandier  and  G.  Wertheim,  1846).  The 


3  66 


TIMBER  IN   TENSION. 


[Art.  38, 


TABLE   I. 


KIND    OF   WOOD. 


Hornbeam 
Aspen .... 

Alder  

Sycamore . 
Maple 


Oak 


Birch  . 
Beech . 
Ash... 
Elm  .. 
Poplar 

Acacia 


Fir  . 

Pine 


COEFFICIENTS   OF    BLAST. 


Pounds. 
,335,000 
,329,000 
,021,000 
,616,000 
,459,000 
,765,000 

to 

1,214,000 
2,431,000 

to 

1,263,000 
1,450,000 
1,798,000 

to 

1,364,000 
1,436,000 
1,027,000 

to 

901,000 
2,206,000  ) 

to 

2,018,000 
2,218,000 

to 

1,310,000 
1,088,000 


ELASTIC    LIMIT. 


Pounds. 

3,060 
4,380 
2,570 
3,270 
3,870 

3,340 


2,300 
3,300 

2,890 
2,62O 

2,100 
4,540 

3,060 
2,320 


ULTIMATE  TENSILE  RESIST. 


Pounds. 
4,250  . 
10,249  (l) 
6,460  (l) 
8,760  (I) 
5,090  (I) 

8,530 


6,110 
5,o8o 
9,640  (i) 
9,940  (i) 
2,800  (i) 

11,280 

5,940 
3,530 


results  are  means,  and  were  obtained  from  small,  well  seasoned 
rods,  with  cross-sectional  areas  varying  from  0.30  square  inch 
(some  fir  specimens)  to  1.50  square  inches  (one  oak  specimen), 
and  are  given  in  pounds  per  square  inch.  The  results  indi- 
cated thus  "(i),"  belong  to  one  tree  only,  others,  to  several. 

The  limit  of  elasticity  is  that  force  per  square  inch  which 
will  produce  a  permanent  elongation  of  0.00005  of  the  original 
length.  £*>-^T-e^-^r ^^flc^c  2&  £-£t*£i  •£++**/&* *  #7/t&ru*<?<,  ffbp 

These  experimenters  found  that  the  elongations  produced 
by  different  weights  were  composed  of  two  parts,  one  perma- 
nent and  one  elastic  ;  the  latter  being  essentially  proportional 


Art.  38.] 


LASLETTS  EXPERIMENTS. 


367 


TABLE    II. 


KIND   OF  TIMBER. 

EXPERIMENTER. 

SP.   GRAY. 

ULT.   RESIST.  IN 
POUNDS   PER   SQ. 
INCH. 

B 

IN    POUNDS   PER 
SQ.  INCH. 

Oak  English            

Laslett. 

0.858 

3,837 

M 

0.893 

7.571 

Oak   French  

M 

0.976 

8,102 

. 

Oak   Dantzic        

1 

0.838 

4*217 

Oak,  American  White  
Oak,  American,  Baltimore  
Oak,  African  (or  Teak)  
Teak,  Moulmein  

| 

0.969 
0.742 
0.971 
0.777 

7,021 

3^832 
7.052 

3,301 

Iron  Wood,  Burmah  

! 

1  .  176 

1  •134 

91656 

7,100 

VO 

jj 

Greenheart    Guiana    

* 

1.141 

8,8.0 

< 

11 

0.917 

5,558 

<3 

41 

0.765 

' 

Mahogany,  Honduras  
Mahogany,  Mexican  .    

u 

•   0.659 
0.655 

2,998 
3,427 

1 

Eucalyptus,  Australia  : 
Tewart 

" 

i  .  169 

10,284 

Mahogany  
Iron-Bark 

M 

0.996 
1.150 

2,940 
8.377 

•a 

Blue  Gum  
Ash   English           

11 

i  049 
0.750 

6.048 
3,780 

| 

Ash   Canadian                       .... 

M 

0.588 

n 

Beech              

N 

o.  705 

4,8-;3 

Elm   English 

/     u 

0.642 

rs 

Rock  Elm.  Canada  
Hornbeam.  England  
Fir,  Dantzic  
Fir,  Riga.     .                  .... 

u 

0.748 
0.819 
0.603 
0.553 

9,182 
6,405 

3'23i 
4,051 

£ 

e 

8^* 

Fir.  spruce,  Canada  
Larch,  Russia                  .... 

lt 

0.484 
0.649 

3'934 
4,203 

• 
•£ 

Cedar,  Cuba     
Red  pine.  Canada        

11 

0.469 
0.553 

2,870 
2,705 

* 

Yellow  pine.  Canada  ...    
Yellow  pine!  Canada  

u 

o  55* 
0.552 

2-759 
2,259 

v; 
1 

Pitch  pine,  American 

M 

4,666 

1 

Kauri  pine.  New  Zealand  
Georgia  pine.  American  
Locust,  American  . 

Hatfield. 

o  544 

4,040 
a6,ooo 
24,8oo 

> 

White  oak,  American  

M 

1g,5oo 

Spruce,  American  
White  pine,  American.  .....   . 
Hemlock  . 

Ik 



19,5°° 

J2,000 

to  the  load  and  the  former  measurable  even  for  small  loads 
and  variable  not  only  with  the  load  but  also  with  the  time 
during  which  the  load  acted  :  that 'the  coefficient  and  limit  of 
elasticity  augmented  with  the  seasoning,  but  that  the  greatest 
elongation  diminished  under  the  same  circumstances  ;  that  if 
the  coefficient  of  elasticity  and  ultimate  resistance  along  the 


368 


TIMBER  IN    TENSION. 


[Art.  38. 


fibre  be  taken  as  units,  the  coefficients  of  elasticity  along  the 
radius  and  tangent  to  the  tree,  will  be  0.165  and  0.091  respect- 
ively, while  the  ultimate  resistances  in  the  same  directions 
will  be  respectively  0.163  and  0.159,  these  results  being  con- 
sidered averages. 

The  ultimate  tensile  resistances  of  many  woods,  domestic 
and  foreign,  are  given  in  Table  II.,  as  well  as  the  specific  grav- 
ities. 

The  column  "  B"  will  be  explained  hereafter,  in  the  chap- 
ter on  transverse  resistance  or  bending. 


TABLE    III. 


WOOD. 

ULT.    RESIST. 
IN  POUNDS  PER 

ELASTIC  LIMIT 
IN  POUNDS  PER 

COEFFICIENT  OF 
ELASTICITY   IN 
POUNDS  PER 

PER    CENT. 
SIOIS 

OF    EXTEN- 
AT 

SQ.   IN. 

SQ.    IN. 

SQ.    IN. 

Elas.  Limit. 

Fracture. 

White  Pine 

6  880 

3QOO 

1  8^  4.OO 

o  40 

O   7^ 

Yellow  Pine  
Locust 

20,700 
28  030 

-    13,200 
IO  2OO 

240,240 

VT\  8^O 

0.63 

I    IO 

1.65 

I  8* 

Black  Walnut  .  . 
White  Ash 

9,79° 

It  4.00 

5,700 
97OO 

213,520 

206  540 

0.53 

o  78 

0.85 
i  d.8 

White  Oak  

n  210 

8  TOO 

22O  I3O 

O   77 

I  .  ^O 

Live  Oak 

10  310 

6  300 

2<17   ^IO 

o  58 

i  in 

The  table  gives  average  results.  Those  determined  from 
experiments  of  Mr.  Laslett  are  of  English  origin  ("  Timber 
and  Timber  Trees,  Native  and  Foreign,"  by  Thomas  Laslett, 
1875);  the  others  are  from  American  experiments  by  the  late 
R.  G.  Hatfield  ("Transverse  Strains,"  1877).  Mr.  Laslett's 
specimens  were  2  inches  square  in  cross  section,  and  genearlly 


Art.  38.]        TESTS  BY  THURSTON  AND  LAIDLEY.  369 

were  30  inches  long,  while  those  of  Mr.  Hatfield  were  about 
0.35  inch  round. 

It  will  be  observed  that  Mr.  Hatfield  reached  far  higher 
results  than  Mr.  Laslett.  This  disagreement  may  be  due  to 
the  larger  cross-sectional  area  of  the  latter's  specimens,  which 
certainly  brings  his  (Mr.  Laslett's)  results  more  nearly  in 
accordance  with  what  might  be  expected  from  such  pieces  as 
are  ordinarily  used  by  engineers.  Mr.  Hatfield's  specimens 
were  far  too  small  for  technical  purposes. 

Table  III.  is  taken  from  a  paper  "On  the  Strength  of 
American  Timber,"  by  Prof.  R.  H.  Thurston  (Jour.  Frank. 
Inst.,  Oct.,  1879).  The  specimens  were  turned  down  to  about 
0.5  inch  diameter  for  a  length  of  4.00  inches. 

The  small  values  of  the  coefficient  of  elasticity,  as  compared 
with  those  given  in  Table  I.,  are  probably  due  to  the  fact  that 
they  were  found  at  the  elastic  limit.  Smaller  intensities  of 
stress  would  probably  give  much  larger  values. 

Prof.  Thurston  also  states  that  timber  in  tension  takes  a 
permanent  set  however  small  the  intensity  of  stress. 

The  values  given  in  Table  IV.  were  found  by  Col.  Laidley, 
U.  S.  Army,  in  the  Government  machine  at  Watertown,  Mass. 
(Ex.  Doc.  No.  12;  4/th  Congress,  2d  Session).  Two  of  the 
specimens  were  about  0.63  inch  in  diameter,  and  one  1.25 
inches.  All  the  rest  possessed  diameters  of  about  one  inch 
each. 

Such  small  specimens  as  those  of  Hatfield,  Thurston,  and 
Laidley,  which  were  probably  selected,  give  much  larger  results 
than  would  be  found  for  large  pieces  of  ordinary  lumber ;  these 
considerations  are  highly  prejudicial  to  the  technical  value  of 
the  results. 

Far  more  importance  attaches  to  the  matter  of  size  and 
character  of  timber  specimens  than  to  those  of  metallic  ones. 
In  the  latter  there  is  at  least  an  approach  to  homogeneity  of 
material,  which  the  presence  of  knots,  conditions  of  growth, 
seasoning,  and  other  influences  effectually  prevent  in  timber 
24 


370 


TIMBER  IN   TENSION. 


[Art.  38 


specimens.  Hence  it  is  the  more  necessary  to  test  timber  in 
circumstances  of  condition  and  size  as  nearly  identical  as  pos- 
sible with  those  which  attend  its  actual  use. 

TABLE    IV. 
Diameter  of  Test  Specimens,  I  inch. 


NO. 

KIND    OF   WOOD. 

ULTIMATE    RSISTANCE    PER   SQUARE   INCH    IN 
POUNDS. 

NO.   OF 
TESTS. 

Greatest. 

Mean. 

Least. 

I 

2 

3 

4 

6 

8 
9 

10 

ii 

12 
13 
14 
15 

16 

Yellow  Pine  

17,922 

11,299 
17,044 
7,466 
19,400 

15,71-1 
14,650 
22,838 
27,532 
i*,  733 
22,703 
12,133 
20,520 
19,610 

15,478 
I3,8lO 
16,160 
8,916 
14,283 
6,787 

14,313 
11,164 
11,492 
18,682 
24,120 
11,632 
17,410 
10,124 
20,390 
15,995 

I2,o66 

5,300 
1  1,  600 
6,107 
4,586 
7,312 
9,286 
13,885 
18,961 
10,667 
12,670 
7,600 
2O,26o 
12,400 

4 
i 
i 
4 
4 

2 

4 
3 
3 
3 
3 
3 
4 
3 

2 

3 

Oregon  Pine  

Oregon  Spruce 

White  Pine               .    . 

Spruce 

White  Wood  .          .    . 

Gum  Wood 

White  Maple      . 

Black  Walnut  

Red  Birch  

White  Ash  

Brown  Ash  
White  Oak  

Red  Oak  

Yellow  Oak 

Hickory  

All  specimens  were  of  well  seasoned  wood. 


CHAPTER  VI. 

COMPRESSION. 

Art.  39. — Preliminary. 

With  the  exception  of  material  in  the  shape  of  long  col- 
umns, but  few  experiments,  comparatively  speaking,  have  been 
made  upon  the  compressive  resistance  of  constructive  materials. 

Pieces  of  material  subjected  to  compression  are  divided 
into  two  general  classes — "  short  blocks  "  and  "  long  columns  ;  " 
the  first  of  these,  only,  afford  phenomena  of  pure  compression. 

A  "  short  block  "  is  such  a  piece  of  material,  that  if  it  be 
subjected  to  compressive  load  it  will  fail  by  pure  compression. 

On  the  other  hand,  a  long  column  (as  has  been  indicated  in 
Art.  25)  fails  by  combined  compression  and  bending. 

Short  blocks,  only,  will  be  considered  in  the  articles  imme- 
diately succeeding,  while  long  columns  will  be  separately  con- 
sidered further  on. 

The  length  of  a  short  block  is  usually  about  three  times  its 
least  lateral  dimension. 

It  has  already  been  shown  in  Art.  4  that  the  greatest 
shear  in  a  short  block  subjected  to  compression,  will  be  found 
in  planes  making  an  angle  of  45°  with  the  surfaces  of  the 
block  on  which  the  compressive  force  acts,  i.  e.,  with  its  ends. 
If  the  material  is  not  ductile,  this  shear  will  frequently  cause 
wedge-shaped  portions  to  separate  from  the  block.  But  the 
friction  at  these  end  surfaces,  and  in  the  surfaces  of  failure 
will  prevent  those  wedge  portions  shearing  off  at  that  angle. 
In  fact  the  friction  will  cause  the  angle  of  separation  to  be 


372  WROUGHT  IRON  IN  COMPRESSION.  [Art.  40. 

considerably  larger  than  45°  ;  let  it  be  called  a.  Then,  in 
order  that  there  may  be  perfect  freedom  in  failure,  the  length 
of  the  block  must  not  be  less  than  its  least  width  or  breadth 
multiplied  by  2  tan  a.  In  some  cases,  a  has  been  found  to  be 
about  55°,  for  which  value 

2  tan  a  =  2  X   1.43  =  2.86. 

It  was  shown  in  the  first  section  of  Art.  32,  that  the 
"  ultimate  resistance  "  to  tension  is  in  reality  a  mean,  and  not 
the  greatest  intensity  which  the  material  exerts.  The  same 
course  of  reasoning  will  show  that  it  is,  also,  in  general,  im- 
possible to  subject  a  short  block  to  a  uniform  intensity  of 
compression  throughout  its  mass,  and  that  the  "  ultimate  re- 
sistance to  compression  "  is  a  mean,  usually  considerably  less 
than  the  greatest  intensity  which  exists  at  the  centre  of  a 
normal  section.  As  the  inner  portion  will  be  supported  later- 
ally by  that  outside  of  it,  large  blocks  of  brittle  material  may 
give  greater  intensities  of  ultimate  resistance  than  small  ones. 


Art.  40. — Wrought  Iron. 

It  is  difficult  to  fix  the  point  of  failure  of  a  short  block  of 
wrought  iron  or  other  ductile  material.  An  excessive  compres- 
sive  force  causes  the  material  to  increase  very  considerably 
in  lateral  dimensions,  or  to  u  bulge  "  out,  so  that  every  in- 
crease of  compressive  force  simply  produces  an  increased  area 
of  resistance,  while  the  material  never  truly  fails  by  crumbling 
or  shearing  off  in  wedges. 

A  short  block  of  wrought  iron  is  usually  considered  to  fail 
when  its  length  is  shortened  by  five  to  ten  per  cent. 

If  /j  is  any  intensity  of  stress  while  /,  is  the  compressive 
strain,  or  shortening  per  unit  of  length  caused  by  /„  then 
according  to  Eq.  (2)  of  Art.  2,  the  coefficient  of  elasticity  for 
compression  at  the  intensity/!,  will  be 


Art.  40.] 


COFFICIENTS  OF  ELASTICITY. 


373 


This  ratio  is  not  constant  for  all  degrees  of  stress  and  strain, 
though  for  wrought  iron,  within  the  elastic  limit,  the  diver- 
gences from  a  mean  value  are  not  great.  Table  I.  contains 
coefficients  of  elasticity  calculated  by  Prof.  De  Volson  Wood, 
in  the  manner  shown  by  Eq.  (i),  from  the  data  determined  by 
*  Mr.  Eaton  Hodgkinson  and  given  in  his  work  before  cited. 
(See  Prof.  Wood's  "  Treatise  on  the  Resistance  of  Materials  "). 


TABLE   I. 


A*v         ' 

£/j, 

*r 

£/,. 

Er 

Pounds. 

Inch. 

Pounds. 

Inch. 

Pounds. 

5,098 

0.028 

20.796,500 

0.027 

21,864,000 

9,573 

o  052 

21,049,000 

0.047 

23,595,000 

14,058 

0.073 

21,979,000 

0.067 

24,273,000 

16,298 

0.085 

21,343,000 





18,538 

0.096 

22,156,000 

0.089 

24,108,000 

20,778 

o.  107 

22,160,000 

O.  IOO 

24,038,000 

23,018 

0.119 

23,587,000 

o  113 

23,587,000 

25,258 

0.130 

22,095,000 

0.128 

23,679,000 

27,498 

o.  142 

22.111,000 

0.143 

22,259,000 

29,738 

0.152 

21,938,000 

0.163 

21,139,000 

31,978 

0.174 

20,979,000 

0.190 

19,478,000 

The  results  belong  to  two  square  bars,  and  El  is  in  pounds 
per  square  inch.  A  is  the  area  of  cross  section  ;  it  was  1.0506 
square  inches  for  the  first  bar  and  1.0363  square  inches  for 
the  other.  Hence  the  bars  were  about  one  inch  square.  They 
were  also  ten  feet  long  (L  —  10.00  feet)  and  required  lateral 
support  to  be  kept  in  alignment  so  as  to  act  like  short 
blocks. 

The  table  shows  that  the  values  of  £l  increase  with/,,  when 
the  latter  is  small ;  an  opposite  result  was  found  for  tension. 

What  may  be  called  the  elastic  limit   is  found  for  /,  = 


374 


WROUGHT  IRON  IN  COMPRESSION.  [Art.  40. 


30,000.00  pounds  per  square  inch  (nearly).  Hence,  it  is  seen 
that  the  greatest  value  of  E^  is  found  for  /z  equal  to  one-half 
to  two-thirds  the  elastic  limit. 

The  same  general  remarks  in  regard  to  the  elastic  limit, 
which  were  made  in  connection  with  tension,  may  be  also  ap- 
plied to  the  compressive  elastic  limit. 

The  "  Steel  Committee  "  of  British  civil  engineers,  in  1870, 
made  some  experiments  on  twelve  bars  of  Lowmoor  wrought 

TABLE   II. 


POUNDS   PER   SQUARE   INCH    FOR 

Elastic  Limit. 

Coefficient  of  Elasticity. 

Pounds. 
29,800 

Pounds. 
29,091,000 

25,800 

29,091,000 

29,100 
26,200 

28,718,000 
28,000,000 

iron,  1.5  inches  in  diameter  and  120  inches  long.  These  twelve 
experiments  were  divided  into  four  sets  of  three  each,  and  the 
table  gives  the  means  of  each  of  these  sets  or  groups.  The  co- 
efficients are  computed  at  the  elastic  limit.  Judging  from  the 
results  in  Table  I.,  smaller  values  of  p^  would  have  given 
larger  values  of  Ev 

As  a  mean  value,  the  coefficient  of  elasticity  for  wrought 
iron  in  compression  may  be  taken  at  28,000,000  pounds  per 
square  inch.  For  every  ton  (2,000.00  pounds)  of  compression 
per  square  inch,  therefore,  a  piece  of  wrought  iron  will  be 
shortened  by  an  amount  equal  to 


Art.  40.] 


ULTIMATE  RESISTANCE. 


375 


2,000 


28,OOO,OOO          I4,OOO 


of  its  length, 


Table  TIL  contains  the  results  of  some  experiments  made 
by  Mr.  Kirkaldy  on  some  specimens  of  Swedish  iron,  in  1866. 
The  last  column  gives  the  per  cent,  of  compression  of  original 
length  which  the  piece  suffered  at  the  point  called  the  "  ulti- 
mate compressive  resistance."  The  results  show  well  the  great 
increase  of  resistance  which  a  short  block  of  ductile  material 
offers  with  the  increase  of  compression. 


TABLE   III. 


SECTION   OF 
SPECIMEN. 

LENGTH. 

POUNDS   PER   SQUARE   INCH   FOR 

PFR   CENT. 
COMPRESSION. 

Elas.  Lira. 

Ult.  Resist. 

In. 
1-5    O 

Ins. 

1.5 

Lbs. 
24,050 

Lbs. 

148,800 

45 

15    0 

1-5 

21,200 

28,100 

4 

i-5    O 

3 

23,300 

84,900 

33 

i.o  n 

i 



184,100 

53 

Table  IV.  gives  the  results  of  experiments  on  some  very 
short  lengths  of  Phoenix  and  Keystone  columns.  The  first  six 
results  are  for  Phoenix  sections  from  experiments  by  the  Phce- 
nix  Iron  Co.,  in  1873  ;  the  two  following  are  for  the  same  sec- 
tion from  experiments  made  at  Watertown,  Mass.,  in  1879; 
while  the  last  result  belongs  to  a  Keystone  section  experi- 
mented upon  by  Mr.  G.  Bouscaren,  in  1875.  Unfortunately • 
the  amount  of  compression  or  shortening,  in  each  instance, 
was  not  recorded.  % 

Reviewing  the  results  given  in  Tables  II.,  III.  and  IV.,  it  is 
seen  that  the  "  elastic  limit  "  of  wrought  iron  in  compression,  in 


CAST  IRON  IN  COMPRESSION. 


[Art.  41, 


TABLE   IV. 


RATIO   OF   LENGTH   TO 

AREA   OF   SECTION, 

ULT.    RESIST.    IN   POUNDS 

LENGTH,  INCHES. 

DIAMETER    OF   SECTION. 

SQ.    IN. 

PER  SQ.    IN. 

8.00 

8.00 
4.00 
4.00 

1.46 
1.46 
0.92 
0.92 

6.97 
6-97 
5.62 
5.62 

6O,57O.OO 
60,390.00 
65,870.00 
65,87O.OO 

4.00 
4.00 
8.00 

I.OI 
I.OI 
1.  00 

2.92 
2.92 
H.9O 

55^60.00 
57,130.00 

8.00 

1.  00 

1  1  .  go 

57,3OO.OO 

9.00 

1.  12 

14-25 

5I,5OO.OO 

short  blocks,  may  be  taken  from  0.4  to  o.  5  its  ultimate  com- 
pressive  resistance,  while  the  latter  may  be  taken  at  about 
60,000.00  pounds  per  square  inch. 


Art.  41. — Cast  Iron. 

The  irregular  elastic  behavior  of  cast  iron,  as  seen  in  ten- 
sion, will  also  be  discovered  in  compression.  Table  I.  contains 
results  computed  from  the  data  obtained  by  Captain  Rodman 
by  testing  solid  cylinders  10  inches  long  and  1.382  inches  in 
diameter.  The  second  column  belongs  to  a  specimen  cylinder 
taken  from  a  lO-inch  columbiad,  and  the  third  or  last  to  a  trial 
cylinder  of  remelted  Greenwood  and  Salisbury  iron.  Neither 
specimen  can  be  considered  to  possess  a  true  elastic  limit,  but 
what  is  ordinarily  so  termed  may  be  taken  at  about  20,000 
pounds  per  square  inch. 

In  the  first  specimen  the  first  permanent  set  took  place 
at  3,000.00,  and  in  the  second  at  5,000.00  pounds  per  square 
inch. 


Art.  41.] 


COEFFICIENTS  OF  ELASTICITY. 


377 


TABLE   I. 


INTENSITY  OF  STRESS. 

COEFFICIENT  OF  ELASTICITY  IN  POUNDS  PER  SQUARE  INCH. 

1,000 

6,896,600 



2,000 

8,888,900 

33,333,300 

3,000 

9,836,IOO 

18,750,000 

4,000 

10,666,700 

13,793,100 

5,ooo 

10,752,700 

13,888,900 

6,000 

II,32O,8OO 

12,766,000 

7,000 

11,352,100 

13,725,500 

8,000 

11,510,800 

13,559,300 

9,000 

II,92O,5OO 

13,432,800 

10,000 

I2,I2I,2OO 

13,333,300 

11,000 

12,290,500 

13,095,200 

12,000 

12,182,700 

13,186,800 

14,000 

12,444,400 

13,207,500 

l6,OOO 

12,260,500 

12,903.200 

18,000 

11,920,500 

12,857,100 

2O,OOO 

II,695,9OO 

12,578,600 

22,000 

11,253,200 

12,290,500 

26,000 

IO,236,2OO 

II,6O7,2OO 

30,000 

8,596,OOO 

IO,IOI,OOO 

35,ooo 



7,658,600 

40,000 

' 

5,333,300 

For  a  bar  ten   feet  long  and  one  inch  square,  Mr.  Eaton 
Hodgkinson  found  the  following  values : 


GREATEST. 
13,216,000.00 


MEAN. 
12,134,100.00. 


LEAST. 
10,837,100.00 


all  in  pounds  per  square  inch.     The  greatest  value  was  found 
at  2,240,  and  the  least  at  38,080  pounds  per  square  inch. 

Since  the  coefficient  of  elasticity  measures  the  stiffness  of  a 
body,  and  since  the  coefficient  of  elasticity  for  wrought  iron  in 
compression  has  been  seen  to  be  at  least  twice  as  great  as  that 
of  cast  iron  in  the  same  condition,  wrought  iron  is  at  least 
twice  as  stiff,  compressively,  as  cast  metal.  A  bar  of  the  latter 
material  will  be  compressed  by  2,000  pounds  per  square  inch, 
about 


37$  CAST  IRON  IN  COMPRESSION.  [Art.  41. 

2,000  I          *  .    '  , 

of  its  length. 


12,000,000      6,000 

If  /  is  the  length  of  a  bar  in  inches,  W  the  compressive 
stress  in  pounds  per  square  inch,  then  Hodgkinson  found  the 
total  decrement  in  inches  for  lo-feet  Low  Moor  cast-iron  bars 
to  be 


A'  =  7(0.012363359  —  y  0.000152853  —  0.00000000191212  W).  (i) 

and  the  permanent  set,  in  inches : 

0.543  A'2  +  0.0013 (2) 

Major  Wade  tested  a  number  of  specimens  of  cast  iron  of 
different  numbers  of  fusions,  in  order  to  determine  the  ultimate 
compressive  resistance.  His  specimens  were  from  0.5  to  0.6 
inch  in  diameter,  and  from  1.25  to  1.5  inches  (nearly).  The 
results  were  as  follows : 

FUSION.  NO.  OF  EXPS.  GREATEST.  MEAN.  LEAST. 

2d 4 H4,504 99>770 84,529 

3d 2 140,415 139,540 138,666 

2dand3d...  2 169,427 168,589 167.752 

2d 2 140,415 136,868 133,321 

3d i 168,251 168,251 168,251 

2d 5 163,528 154,576 144,141 

3d 4 174,120 167,030 156,863 

All  results  are  in  pounds  per  square  inch. 

As  the  specimens  gave  way,  portions  sheared  off  along 
planes  making  angles  with  the  normal  sections  of  specimens 
varying  from  46°  to  62.5°.  This  is  the  characteristic  compres- 
sive fracture  of  cast  iron. 

The  3d  fusion  iron  gave  the  highest  resistance. 

Mr.  Hodgkinson  ("  Report  of  the  Commissioners  appointed 
to  Inquire  into  the  Application  of  Iron  to  Railway  Purposes," 
1849)  to°k  specimens  of  16  different  kinds  of  British  irons,  0.75 


Art.  41.] 


UL  TIM  A  TE  RESISTANCE. 


379 


inch  in  diameter  and  0.75  and  1.5  inches  long,  with  the  follow- 
ing results  : 


GREATEST. 


117,605 86,284 56,445  pounds  per  square  inch. 

'As  a  rule,  the  short  specimens  gave  from  5  to  10  per  cent, 
greater  resistance  than  the  longer  ones.  From  another  set  of 
experiments  with  22  different  kinds  of  iron  (specimens  0.75 
inch  in  diameter  and  1.5  inches  long)  he  found  : 


GREATEST. 


115,995 84,200 54>76i  pounds  per  square  inch. 

Mr.  Hodgkinson  found  that  the  hardness  and  ultimate 
crushing  resistance  of  thin  castings  were  greatest  near  the 
surface,  but  that  in  thick  castings  the  surface  and  heart  gave 
essentially  the  same  results.  He  also  found  that  thin  castings 
gave  considerably  greater  ultimate  resistance  to  crushing  than 
thick  ones. 

Sir  Wm.  Fairbairn  tested  the  effect  of  remelting  on  "  Eg- 
linton  "  No.  3,  hot-blast  iron  with  the  following  results  : 


REMELTINGS. 


ULT.    RESIST. 


REMELTINGS. 


ULT.  RESIST. 


2 97,66o 

3 92,060 

4 9!,i7o 

5 92,060 

6 92,060 

7 91,620 

8 92,060 

9 123,420 


10 129,250 

u 156,350 

12 163,740 

13 147,840  defec. 

14 214,820 

15 171,810 

16 157,920 

17 

18 197,190 


All  results  are  in  pounds  per  square  inch.  It  is  observed 
that  the  I4th  remelting  gives  the  highest  resistance. 

From  what  precedes,  it  is  seen  that  the  ultimate  compres- 
sive  resistance  of  cast  iron,  in  good  ordinary  castings,  may 
safely  be  taken  from  85,000  to  100,000  pounds  per  square  inch. 


380 


STEEL  IN  COMPRESSION. 


[Art.  42. 


Art.  42.— Steel. 

Table  A.  contains  results  taken  from  Prof.  Woodward's 
history  of  the  St.  Louis  arch.  The  elastic  limit  and  ultimate 
resistance  have  been  given  here,  as  well  as  the  coefficient  of 
elasticity,  in  order  to  avoid  reproduction  hereafter.  The 
column  "  /  -r-  d  "  gives  the  ratio  of  length  over  diameter ;  the 
latter  varied  from  about  a  half  inch  to  1.14  inches.  It  will  be 
seen  from  this  ratio  that  the  specimens,  in  many  cases,  were 
somewhat  longer  than  "  short  blocks."  Prof.  Woodward  gives 
many  other  results  with  yet  greater  ratios  of  /  —  d,  in  some  of 
which  £t  reached  38,000,000. 

TABLE   A. 


MAKERS. 

MARK. 

/•+•</. 

POUNDS  PER  SQ.  INCH, 
AT 

COEFFICIENT    OF 
ELASTICITY",  LBS. 
PER  SQ.  INCH. 

Elastic 
Limit. 

Ultimate 
Resist. 

Am.  Tool  Steel  Co.,  Brooklyn. 
Am.  Tool  Steel  Co.,  Brooklyn. 
Wm.  Butcher  
Am.  Tool  Steel  Co.,  Brooklyn. 
Parks  Bros  ,  Pittsburg  

C'ld  chisels. 
C'ld  chisels. 
No.  6. 
Lathe  No.  4 
Boiler  plate. 

it         ii 
u         u 

Ingot  of 
Chrome      • 
Steel. 

Ingot  of 
Chrome 
Steel. 

Chrome       J 
Steel  plate.     j 

3 
5 
4.2 
5 
5-° 
5-5 
5 
5 
5 
5'1 
5-i 
5-i 
5-i 
4-7 
5-2 
5-3 
i.i 
i 
i 
i 
5 
5-1 

50,600 
50,000 
38,000 
53.000 
34,000 
40,000 
23,000 
27,700 
39,600 
43.520 
37.400 
38,900 
45,400 
39'  7<» 
50,030 

5°>°3° 
50,120 
50,120 

145,000 
148,000 
96,200 
105,000 

20,200,000 
3I,OOO,OOO 

13,514,000 
8,506,000 
13,585,000 
15,450,000 
II,8oo.OOO 
15,370,000 
13,226.000 
14.634,000 
13,937,000 
14.371,000 
16,233,000 

U                U                         11 

134,000 
132,800 
136,490 
122,300 
92,000 

102,000 
IO6,OOO 
80.050 
84,050 

86.200 
78,180 
76,500 
92,470 

U                (I                         U 

U           it                 11 

Wm  Butcher 

Wm.  Butcher  .   .            

Butcher  Steel  Works 

i              i 

i              i 

i              i 

U                       I 

Wm.  Butcher  

Wm  Butcher 

In   1868,  Chief  Engineer  Wm.  H.  Shock,  U.S.N.,   tested 


Art.  42.]  RESISTANCE  AND  ELASTICITY.  381 

specimens  3.5  inches  long  and  half  an  inch  square,  of  Parker 
Bros.  "  Black  Diamond  "  steel  with  the  following  results : 

Normal  untempered  steel:  Ult.  Resist,  from  100,100  to 
112,400  pounds  per  square  inch. 

Heated  to  light  cherry-red  and  plunged  in  oil  at  82°  Fahr.: 
Ult.  Resist,  from  173,200  to  199,200  pounds  per  square  inch. 

Heated  as  before,  and  plunged  in  water  at  79°  Fahr.,  with 
final  temper  (plum-blue)  drawn  on  heated  plate :  Ult.  Resist. 
from  325,400  to  340,800  pounds  per  square  inch. 

Heated  as  before  and  plunged  in  water  at  79°  Fahr.,  and 
tested  at  maximum  hardness:  Ult.  Resist,  from  275,640  to 
400,000  pounds  per  square  inch.  In  each  of  these  cases  there 
were  three  tests. 

The  following  values  (each  is  a  mean  of  8  tests)  were  found 
by  the  United  States  Test  Board,  "  Ex.  Doc.  23,  House  of 
Rep.,  46th  Congress,  second  Session/'  for  small  annealed  speci- 
mens of  tool  steel,  of  about  one  inch  in  length  and  0.715  inch 
in  diameter : 

Ult.  comp.  persq.  in. )  I75,992  \  174.586  ;  183,938  ;    193,413  ;  193,197  ',    174,586  ; 
of  original  section.    )  193,517  ;  174,895  pounds  per  square  inch. 

Final  comp.  per  sq.  [134,717;  127,579;  i49»8Si  ;  139,196;  145,751;  128,834; 
in.  of  final  section.  )  125,126  ;  140,489  pounds  per  square  inch. 

The  final  lengths  varied  from  56  to  89  per  cent,  of  the 
originals. 

Kirkaldy's  "  Experimental  Inquiry  into  the  Mechanical 
Properties  of  Fagersta  Steel,"  1873,  furnish  data  from  which 
may  be  computed  a  series  of  values  of  the  ratio  (£,)  of  stress 
over  strain,  or  coefficient  of  elasticity,  for  different  intensities 
of  stress. 

All  the  specimens  were  cut  from  plates  of  mild  steel  of  the 
thickness  shown  in  the  table,  and  were  100  inches  long  and 
2.25  inches  wide.  They  were  laterally  supported  in  a  trough 
arrangement  designed  by  Mr.  Kirkaldy. 


332 


STEEL   IN  COMPRESSION. 


[Art.  42. 


TABLE   I. 


COEFFICIENTS    OF   ELASTICITY    IN    POUNDS    PER   SQUARE    INCH. 


INTENSITY 

Unannealed. 

Annealed. 

OF  STRESS. 

X  inch  plate. 

%-inch  plate. 

}£-inch  plate. 

%-inch  plate. 

IO.OOO 

58,824,000 

38,462,000 

50,000,000 

34,483,000 

14,000 

56,000,000 

33,333,ooo 

48,276,000 

33,333,ooo 

18,000 

54,545,000 

31,034,000 

45,OOO,OOO 

31,034,000 

22,OOO 

48,889,000 

29,730,000 

38,596,000 

26,190,000 

26,OOO 

45,614,000 

28,261,000 

23,214,000 

19,118,000 

30,000 

42,254,000 

24,000,000 

7,792,000 

2,542,000 

34,ooo 

40,000,000 

3,795,000 

5,2O7,OOO 



38,000 

38,384,000 



4,265,000 



42,000 

35,000,000 



3,717,000 



46,000 

30,872,000 







50,000 

25,000,000 

Below  18,000  pounds  per  square  inch  annealing  does  not 
much  change  the  coefficients  for  the  5/6-inch  specimens,  but 
affects  the  thin  ones  more  decidedly.  In  all  the  specimens  the 
elastic  behavior  is  very  irregular,  and  none  of  them  can  be  said 
to  possess  a  true  elastic  limit.  In  the  unannealed  thin  and 
thick  plates,  the  first  permanent  sets  took  place  at  40,000  and 
20,000  pounds  per  square  inch,  respectively ;  in  the  correspond- 
ing annealed  ones,  at  30,000  and  20,000,  respectively. 

By  referring  to  Table  III.  of  Art.  34,  it  will  be  seen  that 
the  coefficients  for  compression  are  considerably  larger  than 
those  for  tension. 

Table  II.  contains  coefficients  of  elasticity  computed  from 
the  results  of  experiments  made  under  the  supervision  of  the 
"  Steel  Committee  of  Civil  Engineers "  (English).  All  are 
computed  for  the  "  limit  of  elasticity."  The  upper  portion  of 
the  table  belongs  to  round  specimens  1.382  inches  in  diameter 
and  50  inches  (36  diameters)  long,  tested  in  1868  ;  the  lower 


V  oK   T]!E  '< 

UNIVEESITY 


Art.  42.] 


LIMITS  OF  ELASTICITY. 


OIF 


383 


TABLE   II. 


QUALITY. 

GREATEST  E  \. 

MEAN  Er 

LEAST  £1. 

Hammered  ) 
and         \ 
rolled.       ) 

Bessemer. 
Crucible. 

26,130,000 
34,461,000 

35,000,000 
33,939,000 

34,461,000 
31,464,000 

Chisel,      } 
tire,  rod,     >• 
roiled,  etc.  ) 

Bessemer. 
Crucible. 

30,270,000 
31,550,000 

29,091,000 
29,474,000 

28,718,000 
28,000,000 

E\  =  coefficient  of  compressive  elasticity  in  pounds  per  square  inch. 

portion  to  1.5  inch  round  specimens  and  120  inches  long, 
which  were  tested  in  1870.  Table  I.  shows  that  considerably 
different  results  might  be  expected  with  lower  intensities  of 
stress. 

TABLE    III. 


LIMITS   OF   ELASTICITY   IN   POUNDS   PER   SQUARE   INCH. 

LENGTH. 

Greatest. 

Mean. 

Least. 

Inches. 

Bessemer  ;  tires,  ham-  1 
mered,  rolled,  fagot- 
ted,  etc. 

53,520 
52,240 

50,250 
48,540 

42,500 
40,990 

1.38 
2.76 

Diam.  =  1.382  ins.    J 

49,800 

46,700 

40,470 

5-53 

Crucible  ;    hammered,  "j 
rolled,  chisel,  tires, 
rods,  etc. 

60,260 
59.000 

55,760 
53,780 

48,990 
43,99° 

1.38 
2.76 

Diam.  =  1.382  ins.    J 

53,740 

49,860 

42,000 

5-53 

The  upper  Bessemer  results  are  for  a  set  of  18,  and  the 


STEEL  IN  COMPRESSION. 


[Art.  42. 


lower  for  a  set  of  1 1  tests  ;  the  upper  crucible  results  are  for  a 
set  of  II,  and  the  lower  for  a  set  of  20  (?)  tests. 

The  "  limits  of  elasticity  "  of  specimens  of  the  same  steels 
to  which  the  upper  portion  of  Table  II.  belongs  (and  for  the 
same  number  of  experiments)  are  shown  in  Table  III. 

The  following  "  limits  of  elasticity,"  in  pounds  per  square 
inch,  correspond  to  the  lower  portion  of  Table  II. : 


Bessemer 

Crucible , 


47,490. 
60,480. 


39,070. 
52,190. 


35,840  pounds. 
36,290  pounds. 


TABLE   IV. 


ELASTIC    LIMIT   IN    POUNDS    PER   SQUARE    INCH. 


1.2. 

0.9. 

0.6. 

0.3. 

I  diam. 

2       " 

4     " 

8     " 

64,OOO 

63,330 
62,330 
6l,670 

62,670 
58,670 
5S,670 
58,000 

60,000 
57,330 
53,330 
52,670 

39,000 
42,000 
41,000 
40,670 

Means. 

62,833 

59,500 

55,830 

40,670 

ULTIMATE    RESISTANCE   IN    POUNDS   PER   SQUARE   INCH. 

I  diam. 

2       " 

4     " 

8     " 

l69,9IO 
133,330 
102,170 

173,290 
117,560 

95,2io 

I56,OOO 
105,330 
84,830 

121,330 
81,760 
47,5io 

Means. 

135,140 

128,690 

115,387 

83,540 

The  results  of  the  experiments  of  Mr.  Kirkaldy  on  speci- 
mens of  different  grades  of  Fagersta  steel  and  of  various 
lengths  in  terms  of  diameter,  are  given  in  Table  IV.  All  the 


Art.  42.] 


ULTIMATE  RESISTANCE. 


385 


specimens  were  turned  to  1.128  inches  (i  square  inch  area)  in 
diameter,  and  were  of  the  lengths  shown. 

The  numbers  1.2,  0.9,  0.6  and  0.3  were  used  to  indicate  the 
different  grades  of  steel,  the  larger  numbers  belonging  to  the 
higher  steels. 

The  specimens  of  one  diameter  in  length  shortened,  under 
a  load  of  200,000  pounds  per  square  inch,  21,  22,  26  and  48  per 
cent.,  respectively,  for  the  marks  1.2,  0.9,  0.6  and  0.3.  Three 
of  the  "  2  diam."  specimens  failed  by  detrusion,  or  by  portions 
shearing  off  obliquely ;  all  the  others  either  bulged  or  took  a 
skew  form,  though  one  of  the  "  8  diams."  finally  broke. 

Table  V.  contains  the  results  of  Major  Wade's  experiments 

TABLE  V. 


ULT.    RESIST.   IN  POUNDS  PER 

DESCRIPTION. 

LENGTH   OVER   DIAMETER. 

SQUARE   INCH. 

Not  hardened  

2    c.c. 

108,044 

Hardened,  low  temper  

2-47 

354,544 

Hardened,  mean  temper  .  .  . 

2-52 

39^985 

Hardened,  high  temper  .... 

2.48 

372,598 

All  specimens  about  I  inch  long  and  0.4  inch  in  diameter. 

on  specimens  of  cast  steel,  in  1851.     The  results  are  seen  to  be 
very  high. 

A  piece  of  the  Hay  steel  used  by  Gen.  Smith  in  the  Glas- 
gow, Mo.,  bridge,  about  il/£  inches  square  and  3*^  inches  long, 
gave  an  ultimate  compressive  resistance  of  139,350  pounds  per 
square  inch  ("  Annales  des  Pouts  et  Chauss£es,"  Feb.,  1881). 


386 


COPPER,   ETC.,   IN  COMPRESSION.  [Art.  43. 


Art.  43. — Copper,  Tin,  Zinc,  Lead  and  Alloys. 

Table  I.  shows  some  coefficients  of  elasticity  (i.e.,  ratios 
between  stress  and  strain),  computed  from  data  determined  by 
Prof.  Thurston,  and  given  by  him  in  the  "  Trans.  Amer.  Soc. 
of  Civ.  Engrs.,"  Sept.,  1881.  The  gun  bronze  contained  cop- 
per, 89.97  ;  tin,  10.00 ;  flux,  0.03.  The  cast  copper  was  cast 
very  hot. 

TABLE   I. 


STRESS  IN  POUNDS  PER  SQUARE 


COEFFICIENTS   OF   ELASTICITY   IN    POUNDS    PER   SQUARE   INCH. 


INCH. 

Gun  Bronze. 

Cast  Copper. 

I,62O 



1,254,000 

3,260 

3,622,000 

1,415,000 

6,520 

4,075,000 

1,651,000 

9,780 

6,113,000 

1,795,000 

13,040 

6,520,000 

1,824,000 

l6,3OO 

5,433,000 

1,842,000 

19,560 

5,148,000 

1,845,000 

22,820 

3,935,000 

1,735,000 

26,080 

2,308,000 

1,503,000 

29,340 



1,144,000 

32,600 

1,073,000 

815,000 

48,gOO 

463,600 

332,500 

The  ratios  of  stress  over  strain  are  far  from  being  constant. 
Strictly  speaking,  therefore,  there  is  no  elastic  limit  in  either 
case*  In  that  of  the  gun  bronze,  however,  it  may  be  approxi- 
mately taken  at  20,000  pounds  per  square  inch  (Prof.  Thurston 
takes  it  22,820),  and  in  that  of  the  copper  at  25,000  pounds. 
The  test  specimens  were  two  inches  long  and  turned  to  0.625 
inch  in  diameter. 

At  38,000  pounds  per  square  inch  the  gun  bronze  specimen 
was  shortened  about  41  per  cent,  of  its  original  length,  while 
its  diameter  had  become  0.77  inch. 


Art.  43.] 


RESISTANCE  OF  ALLOYS. 


387 


TABLE   II. 


g 

i  «  . 

«  i 

Jj*                  Q 

K  3 

POUNDS  PER   SQUARE   INCH 

<   5 

I'll 

COMPOSITION. 

0    & 

b,    *3    ^ 

%Z% 

CAUSING    A    SHORTENING    OF 

J  a 

5       2 

MANNER     OF 

I] 

f-i            U 
g    0    h 
g    Z    < 

w    W 

FAILURE. 

Copper. 

Tin. 

5% 

16* 

20^ 

g    I    3 

D 

97-83 

1.92 

29,340 

34,000 

46,000 

46,260 

0-37 

34,000 

Flattened, 

95-96 

3.80 

39,200 

42,050 

52,150 

52,150 

0.30 

42,050 

44 

92.07 

7.76 

31,500 

42,000 

65,000 

84,100 

0.45 

42,000 

it 

90.43 

9.50 

32,000 

38,000 

6o,000 

61,930 

0.34 

38,000 

44 

87-15 

12.77 

39,000 

53,ooo 

8o,OOO 

89,640 

0-39 

53,000 

it 

80.99 

18.92 

65,000 

78,000 

103,490 

103,490 

O.2O 

78,000 

44 

76.60 

23-23 

101,040 



114,080 

0.09 

114,080 

Crushed. 

69.90 

29-85 







146,680 

0.04 

146,680 

44 

65.31 

34-47 







84-750 

0.03 

84,750 

44 

61.83 

37-74 







39,"o 

O.O2 

39,"o 

it 

47.72 

51  .00 



i 

• 

84,750 

0.02 

84,750 

44 

44.62 

55.15 







35.850 

6.01 

35,850 

u 

38.83 

60.79 





— 

39,  "o 

0.02 

44 

38-37 

61.32 







29,340 

O.OI 

29,340 

i4 

34.22 

65.80 

19,560 





19,560 

0.06 

19,560 

H 

25.12 

74-  51 

V,930 

17,930 

17,930 

17,930 

0.28 

17,93° 

44 

20.21 
15.12 

79.62 
84.58 

16,300 
6,520 

16,300 
6,520 

16,300 
6,520 

16,300 
9,45o 

0.29 

0.51 

16,300 
6,520 

Flattened. 

11.48 

88.50 

10,100 

10,100 

10,100 

14,020 

0.50 

10,100 

44 

8-57 

9I-39 

6,500 





9,78o 

0.06 

9,780 

tt 

3-72 
0.74 

96.31 
99.02 

6,520 

6,520 

6,520 
6,520 

6,520 
6,520 

9,780 

0.34 

0.36 

9,780 
9,780 

i» 

0.32 
Cast  cc 

99.46 
>pper. 

6,520 
26,000 
33,000 

6,520 
39,000 
45,500 

6,520 
51,  coo 

58:670 

9^780 

74,97° 
78,230 

0.38 

0.45 

°-43 

9,780 

39,000 
45,5oo 

it 

n         * 

34,ooo 

42,000 

58,000 

71,710 

0.32 

42,000 

4* 

n         t 

30,000 

36,000 

50,000 

104,300 

0.52 

36,000 

44 

ti         t 

30,000 

37,ooo 

50,000 

91,270 

0.48 

37,000 

it 

Cast  t  n. 

35,000 
6,032 

48,000 
6,400 

65,000 
6,53° 

97,790 
7,500 

0.41 

0.44 

48,000 
6,400 

4' 

The  copper  specimen  failed  at  71,700  pounds  per  square 
inch,  having  been  shortened  about  one-third  of  its  length. 

The  results  of  a  series  of  tests  by  Prof.  Thurston,  in  con- 
nection with  the  United  States  testing  commission,  are  given 
in  Table  II. ;  they  were  abstracted  from  "  Mechanical  and 
Physical  Properties  of  the  Copper-Tin  Alloys,"  United  States 
Report,  edited  by  Prof.  R.  H.  Thurston,  1879.  All  the  speci- 
mens were  0.625  inch  in  diameter  and  2  inches  long.  Scarcely 
one  of  them  can  be  said  to  possess  an  elastic  limit. 


388 


COPPER,   ETC.,   IN  COMPRESSION. 


[Art.  43 


The  series  of  alloys  presents  some  interesting  results. 
About  the  middle  third  of  the  series  are  seen  to  be  brittle 
compounds  giving  (as  a  rule)  high  ultimate  compressive  resist- 
ances, while  the  other  two-thirds  are  ductile,  and  give  at  the 
copper  end  high  results,  and  low  ones  at  the  tin  end. 

It  will  be  observed  that  Prof.  Thurston  took  the  load  per 
square  inch  which  gave  a  shortening  of  10  per  cent,  of  the 
original  length  as  the  ultimate  resistance  to  crushing  of  the 
ductile  alloys  and  metals,  since  such  materials  cannot  be  said 
to  completely  fail  under  any  pressure,  but  spread  laterally  and 
offer  increased  resistance. 

TABLE   III. 


PER   CENT.    OF 

POUNDS   PER   SQUARE   INCH   FOR 

PER   CENT.  OF 

MANNER 

SHORTENING. 

OF    FAILURE. 

Copper. 

Zinc. 

£i. 

Ult.  Resist. 

96.07 

3-79 

305.500 

29,000 

IO.O 

Flattened. 

90-56 

9.42 

342,100 

30,000 

10.  0 

<  < 

89.80 

10.  06 



29,500 

IO.O 

" 

76.65 

23.08 

656,500 

42,000 

IO.O 

" 

60  94 

38.65 

1,772,500 

75.ooo 

IO.O 

" 

55.15 

44.44 



78,000 

IO.O 

« 

49.66 

50.14 

1,345,500 

117,400 

IO.O 

" 

47.56 

52.28 

I,5OO,OOO 

121,000 

IO.O 

" 

25.77 

73-45 

4,232,800 

110,822 

5.85 

Crushed. 

20.81 

77.63 

2,485,000 

52,152 

2.75 

" 

14.19 

85-10 

897,000 

48,892 

10.8 

" 

10.30 

88.88 

49,000 

IO.O 

Flattened. 

4.35 

94-59 



48,000 

IO.O 

" 

0.00 

IOO.OO 

318,500 

22,000 

IO.O 

it 

Table  III.  contains  the  results  of  Prof.  Thurston's  tests  of 
the  copper-zinc  alloys  made  while  he  was  a  member  of  the 
United  States  Board.  The  data  are  taken  from  "  Ex.  Doc.  23, 
House  of  Representatives,  46th  Congress,  2d  Session."  The 
specimens  were  two  inches  long  and  0.625  inch  in  diameter  of 
circular  cross  section. 


Art.  44.]         COMPRESSIVE  RESISTANCE   OF  GLASS. 


389 


The  values  of  Ej_  (ratios  of  stress  over  strain)  are  computed 
for  about  one-quarter  the  ultimate  resistance.  This  ratio  is  so 
very  variable  for  different  intensities  of  stress  that  these  alloys 
can  scarcely  be  said  to  have  a  proper  "  elastic  limit." 

In  the  "Philosophical  Transactions"  for  1818,  Rennie  gives 
the  following  as  the  results  of  his  experiments  on  0.25  inch 
cubes : 

Fine  yellow  brass  (10  per  cent,  shortening). . .  12,852  pounds  per  square  inch. 
Fine  yellow  brass  (50  per  cent,  shortening). . .  41,216  pounds  per  square  inch. 
Cast  lead .(50  per  cent,  shortening). . .  1,932  pounds  per  square  inch. 


Art.  44. — Glass. 

The  following  results  are  taken  from  Sir  Wm.  Fairbairn's 
"  Useful  Information  for  Engineers,"  second  series.  The 
cylinders  were  about  0.75  inch  in  diameter  and  annealed. 

TABLE   I. 


KIND   OF   GLASS. 

SPECIMEN. 

HEIGHT   OF  SPECIMEN. 

CRUSHING  RESISTANCE, 
LBS.   PER   SQ.  INCH. 

Flint 

Cylinder 

Inch. 

I  .OO 

2*1  4.8o 

I.OO 

•34,8^0 

« 

1.  60 

2O,78O 

(( 

2.O5 

^2  8OO 

Green    

I  -OO 

22,58O 

I.5O 

35,030 

ii 

2.OO 

38,O2O 

Crown                        .    .  . 

I.OO 

23,l8o 

I.5O 

38,830 

Flint    

Cube. 

.15 

I4,24O 

.16 

13,200 

a 

.  IO 

I3,26o 

H 

.  IO 

11,820 

.00 

20,470 

.00 

19,950 

Crown   

0.90 

21,870 

390 


CEMENT,   ETC.,   IN  COMPRESSION. 


[Art.  45- 


It  will  be  observed  that  the  cubes  give  considerably  less 
resistance  than  the  cylinders. 

All  the  glass  was  annealed,  but  Fairbairn  remarks  that  the 
cubes  may  have  been  only  imperfectly  so,  since  they  were  cut 
out  of  the  interior  of  larger  masses,  while  the  cylinders  were 
cut  from  rods  as  they  were  drawn.  The  latter,  also,  thus  re- 
tained their  natural  skins,  which  may  have  increased  their 
resistances. 

At  the  instant  of  failure  the  specimens  were  shattered  into 
a  great  number  of  small  pieces. 


Art.  45. — Cement — Cement  Mortar — Concrete — Artificial  Stones. 

Table  I.  of  Article  37  contains  the  ultimate  compressive 
resistances  of  a  great  number  of  pure  cements,  as  tested  by 
General  Gillmore  under  the  circumstances  related  in  connec- 
tion with  the  table.  The  results  are  given  in  pounds  per 
square  inch. 

TABLE   I. 


THESE   F 

ESULTS  AR 

E    LBS. 

CEMENT, 

SAND, 

LENGTH, 

DIAM., 

MEAN 

P 

ER  SQ.    IN. 

AGE,  IN 

Greatest. 

Mean. 

Least. 

0 
O 
0 

T2 

2.5 

H 

E.  L. 
R. 

Ei. 

1,500,000 

800 
1,320 
800,000 

424 

D2O 
500,000 

h«3  to 

j     J43- 

I 
I 
I 

J 

u 

A 

E.  L. 
R. 
Ei. 

587 
783 
1,910,000 

365 
494 
607,000 

191 
26l 
217,333 

1  117  to 

L  I*1- 

2 
2 
2 

» 

u 

4 

E.  L. 
R. 
Ei. 

.  SJ 

6,633,  33° 

182 
213 
1,285,000 

98 
131 
220,450 

1127  to 
135. 

Table  I.  contains  the  results  of  tests  of  "  Fall  City  "  (Louis- 
ville) cement  and  cement  mortar.  The  tests  were  made  by 
Mr.  Bremermann,  by  direction  of  Capt.  Eads,  during  the  con- 


Art.  45.]  CEMENT  MORTARS.  39 1 

struction  of  the  St.  Louis  bridge.  "  E.  L."  is  the  elastic  limit ; 
"  R."  the  ultimate  resistance  to  compression  ;  and  "  E,"  the  co- 
efficient of  compressive  elasticity,  all  in  pounds  per  square 
inch. 

The  following  results  are  from  the  same  source : 

Akron  Cement. 

4     2i-inch  cubes  ;  I  vol.  cement,  o  vol.  sand  ;  ult.  resist.  =  2,140  Ibs. 

2  "     "         "         2  "         "         I    "         "         "       "       =  1,105     " 

3  "     "        "         *  "         "        I    "        "      '".',."'-      733    " 

2  «     .1        «         2  «         <<        3    -        «         ••       «      =      520    " 

I      "     "        "         i  "         "        2    "        "         "       "      =      240    " 

!        «       «  »  I    "  «  4     »  "  "          "         =       480      " 

"  Fall  City"  Louisville,  Cement. 

3  2^-inch  cubes  ;  I  vol.  cement,  o  vol.  sand  ;  ult.  resist.  =  1,587  Ibs. 
!      '•     "         "         2  '*         "         i    "         "         "       "       =     640    '* 
j       •«      "  "  !    "  "  I     "  "  .        <l        "         =      400     " 

i      "     "        "         2  "         "        3    "        "         "      "       =     240    " 


Louisville  Cement  from  Beach  6°  C0. 

4  2^-inch  cubes  ;  I  vol.  cement,  o  vol.  sand  ;  ull.  resist.  =  1,615  Ibs. 

I  "     "         •'  2  '*         "         I    "         "         "       "       =  1,280    <4 

I  "     "     .    "  i   "         "         i    '«         "         "       "       =      560    " 

1  "     •'         "  2  "         ''        3    "         "         "       ''       =      400    " 

2  "     "        "  i   "         «'        2    "         "         "       "       =      280    " 


Louisville  Cement  from  Hulme  &°  Co. 

2     2^-inch  cubes  ;  i  vol.  cement,  o  vol.  sand  ;  ult.  resist.  =  2,320  Ibs. 

2      "     "        "         I   "         "         i    "         "         "       "       =     740    " 
I       "      "  "  2    "       '    '«          3     "  "  ««        "         =      600     " 

These  ultimate  resistances  are  in  pounds  per  square  inch. 
All  specimens  were  "  moulded  under  hand  pressure  only, 
left  12  days  in  water,  and  exposed  six  months  to  the  air." 


392 


CEMENT,   ETC,,    IN  COMPRESSION. 


[Art.  45. 


TABLE    la. 


DESCRIPTION 

OF    PORTLAND    CEMENT   AND    MORTAR. 

ULT.    RESIST.    IN 
POUNDS   PER.    SQ.    IN. 

Neat  Portland  cemen 

3«7QT 

i  Portland  cement  to 

«? 

2  4QI 

« 

2            " 

D-5 

2  OOJ. 

« 

O            " 

^o 

I  4.^6 

« 

4        " 

S    g 

I   ^"^1 

« 

5       "         

CO 

QCQ 

; 

e  ogS 

I  Portland  cement  to 

I  sand 

W 

o    ^-yg 

2       "      .     . 

W* 

2  "^2 

« 

«.'*,.. 

>3o 

2  156 

« 

4     "    

S  £ 

I  7Q7 

n 

*      " 

vO 

I   HAO 

Neat  Portland  cemen 

t 

5o8d 

i  Portland  cement  to 

I  pit  sand   

t/3 

4e5l 

2        "           

0£ 

364.7 

« 

•3           " 

,20 

2  ^Q^ 

if 

A        " 

s  g 

2  2o8 

« 

e        " 

0 

I  678 

The  results  of  a  large  number  of  experiments  on  the  com- 
pressive  resistance  of  Portland  cement  and  mortar,  at  different 
ages,  by  Mr.  John  Grant,  C.  E.  ("  On  the  Strength  of  Cement," 
1875),  are  given  in  Table  la.  The  specimens  were  made  into 
bricks  9  X  4.25  X  2.75  inches,  and  were  compressed  on  their 
flat  sides  of  9  X  4.25  =  38.25  square  inches  area.  The  results 
are  in  pounds  per  square  inch. 

Table  II.  is  taken  from  the  same  work,  and  shows  the  cir- 
cumstances under  which  Mr.  Grant  made  his  experiments. 
Two  sets  of  blocks  were  made  in  each  case ;  one  set  was  kept 
in  air  for  one  year,  and  the  other  in  water  for  the  same  length 
of  time.  The  cubes  were  then  crushed  with  the  results  shown. 
It  is  to  be  observed  that  the  results  are  pounds  per  square 
foot.  Two  series  of  the  blocks  were  formed  by  compressing 
the  material  in  layers  one  inch  thick ;  the  others  were  not 
compressed. 


Art.  45.]          MORTARS  AND  ARTIFICIAL   STONES. 


393 


TABLE    II. 


S    8    &j 


ULTIMATE  COMPRESSIVE   RESISTANCE   IN   POUNDS   PER   SQUARE   FOOT. 


fc    j  •  i 

S      M      1 

3  I  s 

I  I  s 

Compressed. 

Not  Compressed. 

1     a     * 

° 

Kept  in  air. 

Kept  in 

Kept  in  air. 

Kept  in 

Kept  in  air. 

Kept  in 

water. 

water. 

water. 

(Exceptional.) 

I 

239,680 

381,920 

340,480 

301,060 

268,800 

336,000 

2 

333,760 

358,400 

385,280 

309,120 

344,960 

322,560 

3 

253,120 

258,720 

268,800 

318,080 

215,040 

250,880 

4 

230,720 

243,040 

268,800 

250,880 

250,880 

241,920 

5 

199,360 

222,88o 

219,520 

318,080 

215,040 

210,560 

6 

180,320 

203,840 

182,784 

175,616 

163,072 

152,320 

7 

168,000 

180,320 

147,840 

143,360 

125,440 

112,000 

8 

137,760 

170,240 

120,960 

120,960 

II2,OOO 

98,560 

9 

120,960 

153,440 

107,520 

98,560 

89,600 

80,640 

10 

108,640 

107,520 

94,080 

94,080 

71,680 

62,720 

12"  x  12"  x  12"  blocks. 

6"  x  6"  x  6"  blocks. 

6"  x  6"  x  6"  blocks. 

Table  lla.  is  from  the  same  source  as  Table  I. 

The  concrete  blocks  were  pressed  evenly  on  36  square 
inches  until  failure  took  place. 

The  following  results  for  artificial  stones  are  given  by  Mr. 
Henry  Reid  ("  A  Practical  Treatise  on  Natural  and  Artificial 
Concrete,"  1879): 

A  4-inch  cube  of  Ransome's  "  Siliceous  Stone  "  gave  4,200 
pounds  per  square  inch. 

By  experimenting  with  2-inch  cubes  of  "  Rock  Concrete " 
pipes,  Mr.  Reid  obtained  the  following  results  from  two  series : 

GREATEST.  MEAN.  LEAST. 

4,340 3,454 2,684  pounds  per  sq.  in. 

5,650 4,428 3,401 

5,650 4,763 3,107       "         "     "     " 


394 


CEMENT,   ETC.,   IN  COMPRESSION.  [Art.  45, 


TABLE    lla. 

6"  x  6"  x  6"  Concrete  Blocks. 


1 

VOLS.   SAND. 

VOLS.    AKRON 
CEMENT. 

VOLS.  LOUIS- 
VILLE  CEMENT. 

VOLS.  BROKEN 
LIMESTONE. 

ULT.    RESIST.,  LBS. 
PER  SQ.    IN. 

1 

I 

0 

4 

889 

T: 

I 

0 

4 

1,124 

a 

I 

0 

4 

1,170 

2 

o 

4 

722 

£ 

2 

o 

4 

889 

TO 

2 

0 

4 

T»36i 

w 

I 

O 

I 

4 

1,194 

^ 

2 

O 

I 

4 

950 

| 

2 

O 

I 

4 

640 

p 

2 

o 

I 

4 

890 

»-i 

0.5 

o.  5 

4 

1,170 

•"o    jd 

0.5 

0.5 

4 

,361 

3      'g 

0.5 

0-5 

4 

,445 

8  i 

0.5 

0.5 

4 

,280 

4)        ' 
£      O 

0.5 

0.5 

4 

,250 

I  j 

2 
2 

0.5 
0.5 

0.5 
0.5 

4 

4 

918 

,000 

.^H 
^Q          C3 

2 

0.5 

0-5 

4 

,140 

2 

o-5 

0.5 

4 

,361 

<    £ 

2 

0-5 

0-5 

4 

611 

0 

2 

0-5 

0.5 

4 

i,  in 

There  were  six  experiments  in  each  series. 
Three-inch  cubes  of  Victoria  stone  (six  experiments  in  the 
first  series,  and  ten  in  the  second)  gave : 


5,179 

4,708 


MEAN. 
4,422. 

3,955- 


3,294  pounds  per  sq.  in. 

3,578  pounds  per  sq.  in. 


These  cubes  were  made  in  February,  1879,  an<^  broken  in 
May  of  the  same  year. 

"  Two-inch  cubes  of  silicated  stone  made  with  3  parts 
Thames  ballast  and  I  of  Portland  cement,  gauged  with  water, 
and  put  in  the  silicate  bath  for  n  days,  about  12  months  old 
fractured  as  follows : 


Art.  45.]  ARTIFICIAL   STONES.  395 

No.  1 4»237  pounds  per  square  inch. 

No.  2 5.650  pounds  per  square  inch." 

Four  " granitic  breccia"  cubes,  3"  x  3",  about  25  years  old, 
gave  the  following  results  : 


GREATEST. 


8,886 8,028 7,533  pounds  per  square  inch. 

Seven  blocks  of  Sorel  stone,  varying  from  ij£  x  ij^  x  i 
inch  to  2  x  2l/$  x  \y%  inches  gave: 

AGE.  INERT  MATERIAL.  ULT.   RESIST. 

i  year Coral  sand 6,240  Ibs.  per  sq.  in. 

1  "    Pulverized  quartz 7,270    "          "      " 

2  years Washed  flour  of  emery ^,640    "          «       « 

3  "    Fine  marble 11,560    "          "      " 

9  months Mill  sweepings 6,130    "          "      " 

2  years Marble  and  sand 4, 920    " 

Not  known Marble  with  colored  veneer 7,680   "         "       " 

The  weight  of  the  oxide  of  magnesium  varied  from  12  to 
15  per  cent,  of  the  whole. 

The  results  of  a  series  of  tests,  by  Gen.  Gillmore,  in  1870 
and  1871,  on  coignet  b£ton  blocks,  3.5  x  5.5  x  3  inches,  are 
given  in  Table  III.  Two  blocks  of  each  kind  were  tested.  All 
the  blocks  were  two  months  old.  The  results  are  in  pounds 
per  square  inch. 

With  four  2-inch  cubes  of  Frear  stone,  Gen.  Gillmore  ob- 
tained the  following  results  : 

Four  weeks  old 4, 500  pounds  per  square  inch. 

Four  weeks  old 4,626       "         "       "         " 

Three  weeks  old 2,250    .  "         "       "         " 

Sixmonthsold 2,000       "         "       "         " 

These  blocks  were  composed  of  one  measure  of  hydraulic 
cement,  two  and  a  half  of  sand,  moistened  with  an  alkaline 
solution  of  gum  shellac  of  sufficient  strength  to  furnish  one 


BRICKS  IN  COMPRESSION. 


[Art.  46. 


TABLE   III. 


PROPORTIONS    BY  VOLUME,    LOOSELY   MEASURED. 


Cement,  I     common  lime  powder, 

0.4 

;  sand,  5 

.6.. 

935 

831 

«                    it                     « 

r»   R 

tl 

f. 

805 

1 

o  .  o 

>             5 

.0.  . 

987 

•d 

«<        _           <  <           n         « 

•       *  '     <-i 

416 

I 

o.  4 

\             7 

•  5  •  • 

g 

519 

1 

«        j           .«           tt         tt 

0.8 

tt      7   ^ 

551 

PH 

»               /  •  o  •  • 

571 

r* 

(( 

0.4 

;     "   5 

.6) 

649 

1 

i 

gravel 

and 

pebbles, 

5    f 

681 

«        T           ««           «<         « 

0.4 

;  sand,  5 

.61 

675 

1 

i 

gravel 

and 

pebbles, 

13  f 

831 

«        j           «           «         tt    . 

0.8 

;  sand,  5 

.6> 

649 

gravel 

and 

pebbles, 

5    f 

623 

«                    <i           «<         «    ' 

0.8 

;  sand,  5 

16) 

649 

X 

gravel 

and 

pebbles, 

13  f 

753 

COMP.    RESIST.    IN 
POUNDS   PER   SQ.    IN. 


ounce  of  the  shellac  to  I  cubic  foot  of  the  finished  stone. 
Portland  cement  was  used  in  the  first  three  blocks  and  Louis- 
ville cement  in  the  last. 

Specimens  of  artificial  stone  made  under  the  Van  Derburgh 
system  and  used  in  the  walls  of  the  Howard  University  and 
Hospital  buildings  at  Washington,  D.  C.,  in  1868  and  1869, 
varying  in  age  from  3  to  1 6  months,  gave  resistances  of  173  (4 
months  old)  to  564  (10  months  old)  pounds  per  square  inch. 
Another  specimen  of  the  same  stone,  ten  years  old,  gave  1,45  5 
pounds  per  square  inch. 


Art.  46.— Bricks. 

The  first  set  of  results  given  below  are  computed  from  data 
given  by  Gen.  George  S.  Greene,  Jr.,  C.  E.,  in  Vol.  II.  of  the 
"  Trans.  Am.  Soc.  of  Civ.  Engrs." 


Art.  46.]  RESISTANCE   OF  BRICKS.  397 

Nos.  I,  2  and  4  cracked,  but  did  not  crush  to  pieces,  as  the 
others  did. 

NO.  SIZE   OF   BRICK.  SURFACE.  COMP.    RESIST. 

Inches.  Sq.  ins.  Lbs.  per  sq.  in. 

1 2.3      X   3.52   X  4.4 15.5 3,230 

2 2.24  x   3.5      x  4.46 15.6 3,360 

3 '.  2.34  x  3.5     x  4.52 15.8. 2,750 

4 2.34  x  3-46  x  4.46 15.4 1,994 

5 2.30  x  3.46  x  4.50 15.6 2,050 

6 2.28  x  3.46  x  4.60 15-9 2,920 

The  pressure  was  applied  on  the  two  opposite  largest  faces 
of  the  bricks,  giving  blocks  whose  heights  were  only  0.7  their 
least  widths. 

In  Vol.  VII.  of  the  "  Trans.  Am.  Soc.  of  Civ.  Engrs."  Mr. 
Francis  Collingwood,  C.  E.,  gives  the  following  as  the  results 
of  compressing  ten  whole  bricks  on  end : 

GREATEST.  MEAN.  LEAST. 

3,060 2,065 1,524  pounds  per  square  inch. 

For  ten  half  bricks  on  small  side  : 

6,400 4,610 2,900  pounds  per  square  inch. 

For  ten  half  bricks  on  flat  side  : 

4,150 3,370 2,670  pounds  per  square  inch. 

In  regard  to  these  tests  Mr.  Collingwood  says,  "  The 
bricks  were  selected  to  give  a  fair  average  of  '  good  Haver- 
straw  stock  brick/  not  the  hardest  burned.*  No  packing  was 
inserted  in  the  machine  between  the  bricks  and  the  com- 
pressing surfaces  ;  so  that  the  strength  in  compression  repre- 
sents the  case  of  imperfect  beds,  etc.,  although  it  was  found 
that  it  made  but  little  difference."  He  attributes  the  higher 
values  for  the  "  ten  half  bricks  on  small  sides,"  over  those  be- 


398 


NA  TURAL   STONES  IN  COMPRESSION.         [Art.  47. 


longing  to  the   "  half  bricks  on  flat  side,"  to  the  imperfect 
bearing  surfaces  of  the  latter. 

Art.  47. — Natural  Building  Stones. 

The  ultimate  resistances  and  coefficients  of  elasticity  given 
in  Table  I.  were  determined  in  connection  with  the  construc- 

TABLE   I. 


LENGTH  IN 

DIAM.   IN 

POUNDS   P 

ER   SQ.    IN.    FOR 

MATERIAL. 

INCHES. 

INCHES. 

Ultimate 
Resistance. 

£u 

Grafton  Marmesian  limestone  

6  46 

I    14. 

7  2OO 

10,500  ooo 

e,  87 

1.  06 

8  «;oo 

8,400,000 

..      .. 

5-96 

5QO 

1.  06 

I    O7 

2,000 
6  ooo 

8,500,000 
6  000,000 

•5     OO 

•7      X      *? 

ic  400 

8  oo 

2.^8 

IO  IOO 

12,000,000 

I*?    OO 

1  .  11 

10  800 

5,000,000 

Portland  granite 

5  88 

2    l6 

16  ooo 

5  500  ooo 

«             « 

e    oS 

2    l6 

1  8  «;oo 

6  400  ooo 

<             < 

c    07 

2    IS 

17  ooo 

5,000  ooo 

Richmond.    '          ...        

6.OO 

2    1O 

16,400 

13  500,000 

Portland       '             

3.OO 

1    X    3 

11,700 

Missouri  red  granite  

•3.  oo 

3    X    ^ 

12,700 

<«          «         « 

•2     OO 

<2      X      3 

13  ooo 

«         «         (i 

•j.QO 

•2     X     ^ 

12,700 

«         «         «< 

3.00 

^    X    ^ 

13,600 

tion  of  the  St.  Louis  arch,  and  have  been  taken  from  Prof. 
Woodward's  history.  The  following  results  for  Missouri  stones 
are  from  the  same  source  : 


x  3" 
x  3" 

x 


x  3 
x   3" 
x  41 


x  3-rV  x 


ULT.    RESIST. 

cube  brown  ochre  marble  ..........  15,000  Ibs.  per  sq.  in. 

"     sandstone  from  Ste.  Genevieve..  5,330    "     "     "     " 

"             "           "       «             "  5,500    :<     «     «     « 

"            "          "       "            "         ••  3,400    "     "     "     " 


Art.  47.] 


BUILDING   STOiVES. 


399 


TABLE  II. 
Two-inch    Cubes. 


KIND. 

LOCALITY. 

POSITION. 

COMP.  RESIST.  LBS. 
PER  SQ.  INCH. 

POUNDS  I'ER  CUBIC 
FOOT. 

REMARKS. 

Blue  ..  .. 

Staten  Island  NY         1 

Bed. 

22,250 

178  8 

Cracked  before  bursting 

Dix  Island,  Me  

15,000 

166.5 

Burst  suddenly. 

Dark  .  . 

Quincy,  Mass         . 

17,750 

166.2 

Cracked  before  bursting 

Light 

Outncy,  Mass 

14,750 

168  7 

Cracked  before  bursting 

Flagging  .... 

North  River          ..   .. 

13,425 

168.1 

Broke  suddenly. 

Old  Quarry.. 

Westerly,  R  I 

17.750 

165.6 

Old  Quarry.. 

Westerly,  R.  I  

17,250 

165.6 

It                      tt 

Up  River.... 
Up  River  
Niantic  River 
Niantic  River 
Porter's  Rock 
Porter's  Rock 
Gray  

Richmond,  Va  
Richmond.  Va  .  
New  London,  Conn  .  .  . 
New  London,  Conn  ... 
Mystic  River,  Conn  ... 
Mystic  River,  Conn  ... 
Westerly,  R.  I... 

8 
'1 

ET 

Edge. 
Bed. 

21,250 
2O,OOO 
12,500 

H,I75 
18,125 
22.250 
14.687 

166.3 
166.3 
164.4 
164.4 
166.9 

u 
It 

Gray  

Westerly,  R.  I  . 

Edge. 

14,937 

166.9 

M 

Gray  
Gray  
Gray  

Richmond,  Va  
Richmond,  Va.  
New  Haven,  Conn  

Bed. 
Bed. 
Edge. 

I4,IOO 
13^75 
7,750 

164.4 
164.4 
162.5 

It 

it                 it 

Waxy-looking. 

Gray  
Gneiss 

New  Haven,  Conn  
Sachemshead  Quarry, 

Bed. 

9^500 

162.5 

Broke  suddenly. 

Conn  

Edge. 

15.937 

163.7 

tt            it 

Gneiss  

Sachemshead  Quarry, 
Conn  

Bed 

14,000 

163.7 

tt            tt 

Dark 

Duluth,  Minn. 

Bed 

17,750 

*73.7 

Syenitic. 

Dark  
Bluish  -gray  .  . 
Gray  
Glen's  Falls.. 
Glen's  Falls.. 
Lake  

Huron  Island,  Mich... 
Keene,  N.  H  
Pompton,  N.  J  
Glen^s  Falls,  N.  Y  , 
Glen's  Falls,  N.  Y..   .. 
Lake  Champlain,  N.  Y. 

Bed. 
Bed. 
Bed. 
Bed. 

ssr- 

18,125 
io,375 
24,040 

*M75 
10,750 
25,000 

164.4 
166.0 

168.8 
168.8 
171.9 

Average  of  3. 
Burst  without  cracking. 

u            t             tt 

Lake 

Lake  Champlain,  N.  Y. 

Edee 

21,500 

171  .Q 

tt            t            tt 

North  River. 

Kingston,  N.  Y  

BeX 

13,900 

168.2 

u                 t                  tt 

North  River. 
White 

Kingston.  N.  Y  
joliet    III     .. 

sr 

11.050 
12,775 

168.2 

158  8 

u               t                tt 

White  
Drab  
Drab  
Drab  
Drab 

Joliet,  111  
Lime  Island,  Mich  
Lime  Island.  Mich  
Marquctte,  Mich  

Limcston 

Bed. 
Bed. 
Bed. 
Bed. 
Fdere 

16.900 
25.000 
'5-425 
7-825 
7  600 

162.5 
161.2 

159-4 
146.3 
146  3 

((               t                tt 

Rather  a  clay  stone. 

Dark  
Dark  .. 
Drab  

Bardstown,  Ky  
Bardstown,  Ky  
Canton    Mo  

Bel 
Edge. 
Bed. 

16.250 
15.000 
9.250 

166.9 
166.9 
146.0 

Drab 

Bed 

•  5-650 

146.0 

Caen  
Caen 

France  :  
France                   .  .       . 

Bed. 
Bed. 

3  :65° 
3-45° 

118.8 
118.8 

East  Chester 

Tuckahoe  NY.          "] 

Bed 

12.950 

179  7 

East  Chester. 
Vermont  .  ... 
Vermont 

Tuckahoe.  N.Y  
Dorset.  Vt  
Dorset  Vt        .... 

3 

'la 

Bed. 
Bed. 
Bed. 

12.050 
7.612 
8.670 

179-7 
164.7 
167.8 

400 


NATURAL   STONES  IN  COMPRESSION.         [Art.  47. 


TABLE    II.— Continued. 


y 

• 

3 

•J           ^ 

D 

j 

H'           5 

U 

KIND. 

LOCALITY. 

H 

1     6- 

g  s* 

REMARKS. 

I 

K         (f. 

s    £ 

U5          2 
Q 
2 
D 

8 

2 

Drab  .. 

Mill  Creek  Quarry,  111  . 

Bed. 

9,687 

160.6 

Drab 

Mill  Creek  Quarry,  111 

w 

Edge 

9.787 

156.9 

Drab  .. 
Drab  
Common  Ital. 

North  Bay  Quarry,  Wis. 
North  Bay  Quarry,  Wis. 
Italy  .  . 

3 

3 

Bed 
Edge. 
Bed. 

20,025 
13,700 
11,250 

168  ,'° 

Common  Ital. 

Italy  

Bed. 

13.062 

168.2 

Brown  
Brown  

Little  Falls,  N.  Y  
Little  Falls.  N.  Y 

Bed. 
Edge. 

9.850 

9^5° 

140.6! 
140.6) 

Broke  sud'ly.    Hardened 
by  years  of  exposure. 

Gray  

Belleville,  N.  T  

Bed. 

11.700 

141.0 

Gray  

Belleville,  N.  J  

Edge. 

10,250 

141.0 

Brown  

Middletown,  Conn  

Bed. 

6:950 

148.5 

Brown  

Middletown.  Conn  .... 

Edge 

5)55° 

148.5 

Pink  

Medina,  N.  Y... 

Bed 

17,250 

150.6 

Pink  

Medina,  N  Y 

Edge. 

14.812 

Drab 

Berea,   Ohio  

Bed. 

10,2^0 

III    O 

Drab  

Berea,  Ohio 

Bed. 

8,300 

1*3  J 

Drab  

Berea,  Ohio  

Bed. 

7.250 

137.5 

Drab  

VermiU  ion.  Ohio  

Bed. 

8,250 

135  3 

Drab  

Vermillion,  Ohio     .... 

Bed. 

6,000 

j  .,  e  .  o 

Purple  

Fond  du  Lac,  Wis.... 

Bed. 

6,250 

138^8 

Purple  

Fond  du  Lac.  Wis  

Edge. 

138.8 

Purple  
Purple  
Red-brown  .  . 

Marquette,    Mich  
Marquette,    Mich  
Seneca  Freestone,  O... 

<u 

.1 

|ge. 

7^45° 
5,73° 
9,687 

149.3 

Red-brown  .  . 

Seneca  Freestone,  O.  .. 

"a 

Edge. 

10,500 

Olive  green  .  . 

Cleveland,  Ohio  

$ 

Bed. 

6,800 

I4O.O 

Olive  green.. 

Cleveland,  Ohio..     .  . 

Edge. 

7.910 

I4O.O 

Brown  

Albion,  N.  Y  .  . 

Bed. 

13.500 

151.2 

Brown  
Pink  

Albion,  N.  Y  
Kasota,  Minn  

Edge. 
Bed. 

",35° 
10.700 

I5I.2 

164  4 

Calcareous. 

Pink 

Kasota,  Minn 

Edge. 

11.675 

Light  buff  .. 

Fontenac,   Minn  

Bed. 

6,250 

J45  3 

Light  buff.  .  . 
Freestone  

Fontenac,   Minn  
Dorch'ter,  New  Bruns- 

Edge. 

7,775 

145-3 

wick  

Bed. 

9,150 



Freestone  

Dorch'ter,  New  Bruns- 

wick   

Edpe 

6,050 



Yellow  drab. 

Massillon,  Ohio  

Bed 

8.750 

131.8 

Yellow  drab. 

Massillon,  Ohio  . 

Bed! 

* 

6.725 

131.8 

Craigleith  .  .  . 
Craigleith.... 

Edinburgh,  Scotland.. 
Edinburgh,  Scotland..  - 

Bed. 

Edge. 

12.000 

11,250 

Mi-3 

Table  II.  and  the  other  tables  of  this  article  contain  the  re- 
sults of  tests  given  in  the  "  Report  on  the  Compressive  Strength, 
Specific  Gravity  and  Ratio  of  Absorption  of  the  Building 
Stones  in  the  United  States,"  by  Gen.  Q.  A.  Gillmore,  1876. 


Art.  47.] 


SANDSTONE   CUBES. 


4OT 


The  specimens,  whose  tests  are  given  in  Table  II.,  were  2-inch 
cubes.  "  Each  cube  was  placed  between  two  cushion  blocks  of 
soft  pine  wood,  2  inches  by  2  inches  square,  and  slightly  more 
than  0.25  inch  in  thickness  ;  one  on  the  top  and  the  other 
under  the  bottom  ;  the  grain  of  the  wood  being  parallel  in  each 
to  the  other — though  no  difference  was  observed  when  this 
was  changed,  as  regards  amount  of  record."  ..."  The 
cubes were  brought  to  a  true,  smooth  and  regu- 
lar, but  not  a  polished  surface."  The  third  column  shows 
whether  the  specimen  was  crushed  "  on  bed  "  or  "  on  edge." 


TABLE   III. 
Berea  Sandstone  Cubes. 


EDGE   OF   CUBE. 

COMP.  RESIST.,  LBS.  PER 
SQUARE    INCH. 

EDGE  OF  CUBE. 

COMP.  RESIST.,  LBS.  PER 
SQUARE     INCH. 

Inch. 

O.25 

Pounds. 

4,992 

Inches. 
2.00 

Pounds. 

8,955 

0.50 

6,080 

2.25   . 

9^30 

0-75 

6,347 

2.50 

8,856 

1.  00 

6,990 

2.75 

9,838 

1.25 

7,342 

3.00 

10,125 

1.50 

8,226 

4.00 

11,720 

i-75 

9>3io 

-  — 



General  Gillmore  showed  that  the  size  of  the  cube  tested, 
affected  very  greatly  the  ultimate  compressive  resistance  per 
unit  of  area  of  face  of  cube.  Table  III.  shows  the  results  of 
gradually  increasing  the  size  of  cubes  of  Berea  sandstone, 
crushed  "  on  bed  "  between  wooden  cushion  blocks  increasing 

(with  size  of  cube)  from  about  0.0625  inch  to  about  0.4  inch  in 
26 


402 


NATURAL   STONES  IN  COMPRESSION.         [Art.  47. 


thickness.  The  general  result  is  very  marked  in  spite  of  two 
or  three  irregularities. 

These  results  are  natural  consequences  of  the  character  of 
stone  and  the  cubical  form  of  the  specimens.  A  few  of  Gen- 
eral Gillmore's  experiments  showed  that  such  results  would 
probably  not  appear  if  the  length  of  the  specimens  had  been 
two  or  three  times  the  width  or  breadth. 

The  effect  of  different  bearing  surfaces  on  the  ultimate 
compressive  resistance  of  stone  cubes  is  well  shown  by  the 
results  given  in  Table  IV.  All  the  results  are  in  pounds  per 
square  inch,  and  belong  to  two-inch  cubes,  with  the  exception 
of  the  "Sandstone,  drab"  specimens,  which  were  1.5  inch 
cubes.  Each  result  is  a  mean  of  two  to  five  tests. 

TABLE   IV. 


ULT.    COMP.    RESIST.,    POUNDS     PER   SQUARE   INCH. 


Steel. 

Wood. 

Lead. 

Leather. 

Granite,  Millstone  Point,  Conn  
Granite,  Keene,    N.  H  
Marble    East  Chester    NY.... 

23,190 
24,000 

I  Q  12^ 

22,830 
19,830 
1  7  ^J.O 

15,730 
14.480 
II  560 

15,730 

Sandstone,  Berea    Ohio     .  .          ... 

II   26O 

10  290 

7  080 

6  7^0 

Vermont  marble    Vt   

13  28O 

10  850 

Q  2OO 

8  190 

I  O7s; 

1  O7^ 

I  07$ 

I  075 

Sandstone    drab        .                  . 

4  ooo 

4  ooo 

4  ooo 

Sandstone,  Massillon,  Ohio  
Sandstone,  Massillon,  Ohio  (softer). 

8,500 
5,66o 

8,750 
6,730 

7,250 
5,500 

3,640 

The  steel  cushion  gave  the  highest  results  by  a  little.  A 
soft  cushion  seems  to  be  driven  into  the  small  cavities  and  in- 
terstices of  the  specimen,  and  thus  to  produce  a  splitting 
action  at  the  bearing  surfaces.  "  The  beds  of  the  granite  and 
marble  cubes  were  rubbed  to  the  border  of  polish  ;  those  of 
sandstone  were  rubbed  smooth." 


Art.  48.] 


TIMBER. 


403 


Again,  polished  and  unpolished  cubes  give  different  resist- 
ances per  square  inch,  as  shown  in  Table  V.  The  results  there 
given  are  for  two-inch  cubes  pressed  upon  by  wooden  cushions. 

It  is  at  once  evident  that  the  polished  cubes  gave  consid- 
erably the  highest  resistances.  This  is  probably  due  to  the 
fact  that  the  splitting  action  of  the  wooden  cushions  was  re- 
duced to  a  minimum  on  the  polished  surfaces. 


TABLE  V. 


ULT.    COMP.    RESIST. 

I'ER   SQUARE   INCH. 

Polished. 

Unpolished. 

Granite    Ouincy    Mass                                . 

Pounds. 
2  i  7CO 

Pounds. 

17  7<O 

Granite    Staten  Island    N    Y     

25  ooo 

22  250 

21  630 

XT   -7§O 

Granite    Tarrytown    N    Y 

27  7CO 

18  2^0 

Granite    Millstone  Point    Conn 

22  880 

18  7^0 

Granite    Keene    N  H     

IQ  830 

12  7^O 

Granite,  Westerly,   R.   I   

2-J  CQO 

17  7">O 

Marble,  East  Chester,  N.  Y  

17  «;4,O 

12  Q«;O 

Marble    Vermont  Vt  

10  850 

8  7co 

General  Gillmore's  experiments  show,  in  a  very  conclusive 
manner,  that  variety  in  circumstances  of  testing  will  produce 
a  variety  of  results  for  the  same  section  of  stone  specimen. 
Attending  circumstances  and  dimensions  of  specimens,  there- 
fore, should  always  be  given. 


Art.  48.— Timber. 

Table  I.  is  based  upon  results  of  experiments  made  at  the 
Stevens  Institute,  which  were  given  by  Prof.  Thurston  in  the 
Journal  of  the  Franklin  Institute  for  Oct.,  1879.  The  speci- 
mens were  well  seasoned  and  turned  to  about  1.125  inches  in 


404 


TIMBER  IN  COMPRESSION. 


[Art.  48. 


TABLE    I. 


WOOD. 

POUNDS  OF  STRESS   PER   SQUARE  INCH  AT 

PER    CENT.   OF 
FINAL   SHORTENING. 

Ult.  Resist. 

Elas.  Lim. 

Coefficient  of  Elas. 

White  pine  .... 

Q.  5QO 

^  600 

VCA    AQO 

3e 

Yellow  pine  

H,950 

7,000 

469,8OO 

2.9 

Locust  

14,820 

9  800 

6O4.  Q^O 

3<j 

Black  walnut.  .  .  . 

7,000 

5,700 

1,079,500 

1-25 

White  ash  .  

8,150 

5,  1  80 

713,300 

2-3 

White  oak 

7,140 

5,600 

361,300 

3-3 

Live  oak       .    . 

10,410 

6,300 

594,350 

3-4 

TABLE    II. 


WOOD. 

NO.    OF   EX- 
PERIMENTS. 

ULT.   RESIST.    IN   POUNDS   PER  SQUARE   INCH. 

Greatest. 

Mean. 

Least. 

9 
9 
9 
9 
9 
9 

11,500 
7,500 
12,580 
9,780 
8,410 
6,280 

9»520 
6,640 
11,720 
8,000 
7,860 
5,690 

8,170 
5,880 
II,OIO 

6,530 
7,170 

5,210 

AVhite  pine          

White  oak      

Hemlock          

Art.  48.] 


LASLETT'S  EXPERIMENTS. 


405 


diameter  with  a  length  of  2.25  inches  ;  they  were  compressed 
in  the  direction  of  the  fibre.  The  coefficients  of  elasticity  were 
computed  at  the  "  elastic  limit,"  i.  e.y  at  the  point  at  which 
permanent  set  began. 

Table  II.  contains  the  results  of  experiments .  made  by 
R.  G.  Hatfield  ("Transverse  Strains,"  1877).  The  specimens 
were  from  one  to  two  diameters  high,  and  were  compressed  in 
the  direction  of  the  fibres. 

The  mean  results  of  numerous  English  experiments  by 
Thomas  Laslett  ("  Timber  and  Timber  Trees,  Native  and 
Foreign,"  1875)  are  given  in  Table  III.  He  found  very  little 
difference  in  the  results  for  i-inch,  2-inch,  3-inch  and  4-inch 
cubes ;  those  for  the  smaller  cubes,  as  a  rule,  gave  a  slight 
excess  over  the  others.  The  cubes  were  crushed  in  the  direc- 
tion of  the  fibre. 

TABLE   III. 


TIMBER, 

i,  2,  3  and  4-inch  Cubes. 

ULT.  RESIST.  IN 
LBS.  PER  SQ.  IN. 

TIMBER, 

i,  2,  3  and  4-inch  Cubes. 

ULT.  RESIST.  IN 
LBS.  PER  SQ.  IN. 

Oak,  English  (unseasoned). 
Oak,  English  (seasoned).  .  .  . 
Oak,  French  

4.900 
7.480 

i 
!  Mahogany,  Mexican  
Eucalyptus,  Tewart  
Eucalyptus   mahogany    . 

5,600 
9'350 
7,1?0 

Oak,  Tuscan  
Oak,  Sardinian  
Oak,  Dantzic  

5-47° 
5,835 
7  480 

|  Eucalyptus,  iron-bark  
|  Eucalyptus,  blue-gum  
1  Ash    English 

10,300 
6,900 
6,970 

Oak,  Ameiican,  white  

6,070 

Ash,  Canadian  

5,490 

Oak,  American,  Baltimore.. 

5|89O 

Elm,  English 

i  5'78o 

Teak,  Moulmein     .    ... 

Elm  rock 

8  =;8o 

Iron  wood 

8  310 

Chow     

I  Fir,  Dantzic 

Greenheart 

Sabicu  

8,470 

Fir,  spruce  .       . 

4,850 

Mahogany,  Spanish  

6,400 

Larch  

5.820 

Mahogany.  Honduras    

6,580 

Cedar..-.  

4480 

Red  pine  

5,690 

Pitch  pine 

6,470 

Yellow  pine  

6  432 

The  results  of  the  compressive  tests  of  short  blocks  of 
timber,  during  the  construction  of  the  St.  Louis  bridge,  are 
given  in  Table  IV.  These  are  especially  valuable,  both  in 


406 


TIMBER  IN  COMPRESSION. 


[Art.  48. 


consequence   of  the  large  size   of  the  blocks  and  the  fact  that 
the  pressure  was  applied  with  and  across  the  fibre. 

The  blocks  are  seen  to  be  from  two  to  eight  times  as  strong 
with  the  fibre  as  across  it. 


TABLE   IV. 


KIND  OF  TIMBER. 

WITH   OR  PER- 
PENDICULAR 
TO    FIBRE. 

DIMENSIONS 
IN    INCHES. 

t 
o 

2 

H 
2 

ULTIMATE   RESISTANCE   IN 
FOUNDS   PER   SQ.  IN. 

REMARKS. 

Greatest. 

Mean. 

Least. 

White  oak... 

Perp. 
With. 
Perp. 
Perp. 
Perp. 
Perp. 
Perp. 
With. 
Perp.  . 

with.: 

Perp. 
With. 
With. 
With. 
With. 

444 

444 
333 
333 
333 

I       I       I 
666 
666 
666 
666 
666 
666 
666 
666 

4 
4 

2 
2 
2 
2 

3 
3 
3 
3 
3 
3 

2 

2 
2 

2,200 

2.OOO 

9*700 

440 
3.100 
722 

3,361 

1,222 

4,9/7 
4-14 
3-  1(  6 
3,^94 
4,7*2 

1,750 
3,375 
i,  800 

2,550 
610 
3,24i 
1,092 
4,796 
426 

3^91 
4,611 
3,764 

1,300 
3,200 

i^Soo 

2,000 

555 
3,083 
1,000 
4,722 

417 
3.000 
2,889 
4^5°° 
3,75° 

|  Not    well 
)  seasoned. 

. 

F 

White  oak  
Black  oak 

Gum  

Cvoress 

Ash 

White  pine  
White  pine  
Yellow  pine  ...  . 
Yellow  pine  ...  . 
Cypress  
Cypress 

White  pine  
Yellow  pine  ...  . 
White  oak  

Table  V.  contains  the  results  of  tests  by  Colonel  Laidley, 
U.S.A.,  "  Ex.  Doc.  No.  12,  4/th  Congress,  2d  Session."  A 
few  other  tests  of  short  blocks  from  the  same  source  will  be 
found  in  the  article  on  "  Timber  Columns."  Unless  otherwise 
stated,  all  the  specimens  were  thoroughly  seasoned. 

In  this  table,  the  "  length  "  of  all  those  pieces  which  were 
compressed  in  a  direction  perpendicular  to  the  grain  might, 
with  greater  propriety,  ^e  called  the  thickness,  since  it  is 
measured  across  the  grain. 

In  the  tests  (24-60),  the  compressing  force  was  distributed 
over  only  a  portion  of  the  face  of  the  block  on  which  it  was 
applied ;  thus  the  compressed  area  was  supported,  on  the  face 
of  application,  by  material  about  it  carrying  no  pressure.  In 
some  cases,  this  rectangular  compressed  area  extended  across 


Art.  48.] 


LAIDLEY'S  EXPERIMENTS. 


407 


the  block  in  one  direction  but  not  in  the  other.  In  all  such 
instances  the  ultimate  resistance  was  a  little  less  than  in  those 
in  which  the  area  of  compression  was  supported  on  all  its 
sides. 

TABLE  V. 


.2 

KIND   OF   WOOD. 

LENGTH,  INS. 

COMPRESSED 
SECTION  IN 
INCHES. 

ULT.  RESIST., 
LBS.  PER  SQ. 
INCH. 

PERP.  TO  OR 
WITH  GRAIN. 

REMARKS. 

I 

Oregon  pine  

16.5 

2.46    X    2.O 

8,496 

With. 

2 

Oregon  pine 

IQ    O 

1  .  2  1    X    1  .  2  t 

9,041 

3 

Oregon  pine  ....... 
Oregon  maple 

*y  *y 
ig.g 
8.0 

1.21    X    1.21 

3.63  x   3.63 

8,253 

6,66  1 

>4 

i 

Oregon  spruce  
California  laurel  .  .  . 

24.02 
8.0 

3-9*  x   5.75 
3.58  x  3.58 

5,77-* 
6,734 

, 

Unseasoned. 
Worm-eaten. 

7 

Ava  Mexicana.   .. 

8.0 

3.69  x  3.69 

6,382 

4 

8 
9 

Oregon  ash  
Mexican  white  ma- 

8.0 

3.64  x  3.64 

5,121 

4 

10 

hogany  
Mexican  cedar  

8.0 
8.0 

3-77  x  3.77 
3-75  x  3-75 

4,814 

*k 

ii 

Mexican  mahogany 

8.0 

3-75  x  3.75 

10,043 

44 

I  2 

White  maple 

12.  0 

4  .00  x   4.00 

13 

White  maple  

12.0 

4.00  x   4.00 

?,2IO 

n 

Red  birch  

13.0 

4.26  x  4.26 

,030 

" 

jg 

Red  birch  

13.0 

4.26  x  4.26 

ii 

16 

Whitewood  

12.  O 

4.00  x  4.00 

4,44° 

ii 

17 

Whitewood  

12  .O 

4.00  x  4.00 

4Q3O 

ii 

18 

WMte  ptne 

12.  0 

4  .00  x   4.00 

,jj 

ii 

19 

White  pine. 

12.0 

4.00  x  4.00 

5  760 

M 

20 

White  oak...".  

12.0 

4.00  x   4.00 

7,375 

" 

21 

White  oak  

J2.O 

4.00  x  4.00 

7,010 

ii 

22 

Ash      

12  .O 

4.00  x  4.00 

7  O4O 

u 

23 

Ash  

12.0 

4.00  x   4.00 

/  iw* 

7,640 

" 

24 

Oregon  pine  

1.95 

3.45  x   3.00 

Pcrp. 

25 

Oregon  maple  

3.63  x  3.00 

3jS 

•4 

26 
27 

Oregon  spruce.  .... 
Oregon  spruce.    .  .. 

3-92 

3-92 

5-75  x  4.75 
4.75  x  4.00 

710 
680 

M 

Unseasoned. 
Unseasoned. 

28 

California  laurel... 

3.58 

3.58  x  3.00 

2,000 

44 

29 

Ava  Mexicana  

3-69 

3.69  x   3.00 

2,100 

44 

30 

Oregon  ash  
Mexican  white  ma- 

3-64 

3.64   x  3.00 

2,200 

hogany  

3-77 

3.77  x  3.00 

2,150 

44 

32 

Mexican  cedar 

3-75 

3.75  x  3.00 

I,95° 

44 

33 
34 

Mexican  mahogany 
White  pine  

23 

3-73   x   3-oo 
6.20  x  4.75 

4,500 

875 

M 

White  pine  
Whitewood  

2.90 
3.15 

4-75  x  4-00 
4  75  x   6.20 

1,012        . 
900 

U 

ii 

Mean  of  two. 
Mean  of  two. 

37 

Whitewood    ...    . 

3  15 

4-75  x  4'°° 

I,OOO 

ii 

Mean  of  four. 

38 
39 
4° 

Black  walnut  
Black  walnut  
Black  walnut  

0.875 
0.875 
0.875 

4.75  x  4.00 
4.00  x  3.94 
4  oo  x   2.50 

2,450 

2,200 

2,525 

» 

Mean  of  two. 
Mean  of  two. 
Mean  of  two. 

42 

White  oak  
Spruce  

2.40 

3-70 

4-73  x  4.00 
4.75   x  4.00 

'970 

» 

Mean  of  four. 

43 

Yellow  pine 

3  9^* 

4  .00  x   4  ,00 

i  ooo 

i4 

44 

Black  walnut  

0.75 

4  05   x   4.00 

i.vy^w 

2,800 

44 

45 

bt                      ii 

I.  00 

4.05  x  4.00 

2,560 

4k 

46 



1.25 

4.05   x  4.00 

2,400 

TIMBER  IN  COMPRESSION. 


[Art.  48. 


TABLE  V '.—Continued. 


ri 

2 

COMPRESSED 

ULT.  RESIST., 

£ 

PERP.  TO  OR 

g 

KIND  OF  WOOD. 

0 

SECTION  IN 

LBS.  PER  SQ. 

REMARKS. 

•z 

WITH  GRAIN. 

3 

INCHES. 

INCH. 

47 

Black  Walnut 

•  5° 

4.05    X    4.00 

2,500 

Perp. 

48 

»• 

4.05    X    4.00 

2,400 

* 

49 

«k 

.06 

4.05   x   4.00 

2,360       , 

1 

So 

W 

ite  pine  .  . 

•75 

4.05    x   4.00 

1,1  2O 

1 

51 

lt 

• 

.00 
.25 

4  05   x   4.00 
4  .05   x   4.00 

1,100 

, 

53 

" 

-50 

4.05   x   4.00 

1,070 

• 

54 

14 

•75 

4.05   x  4.00 

i,c6o     . 

55 

4i 

.00 

4.05  x   4  oo 

1,000 

56 

Yellow  birch. 

•25 

4.25   x  3.00 

2,000 

4 

57 

Yellow  birch 

•25 

5.98   x  3.00 

1,650 

4 

58 
59 
60 

W 

W 
W 

lite  maple, 
lite  maple 
lite  oak 

. 

.00 
.00 

$95 

3.95   x   3.00 
5.98  x   3.00 
3.96  x  3.00 

1,700 
1,900 
2,500 

\ 

Mean  of  two. 

The  "  ultimate  resistance  "  was  taken  to  be  that  pressure 
which  caused  an  indentation  of  0.05  inch. 

Nos.  (44-55)  show  the  effect  of  varying  thickness  of  blocks. 
Within  the  limits  of  the  experiments,  the  ultimate  resistance  is 
seen  to  decrease,  somewhat,  as  the  thickness  increases. 

The  results  of  the  experiments  given  in  this  article  show 
conclusively  that  the  ultimate  compressive  resistance  of  short 
blocks  of  timber  depend^  upon  a  number  of  conditions,  such 
as  method  of  compression,  quality  of  material,  size  of  block, 
etc.,  etc.  These  reasons  account  for  the  different  results  ob- 
tained by  different  experimenters  for  the  same  kind  of  timber. 


CHAPTER    VII. 

COMPRESSION. — LONG  COLUMNS. 

Art.  49. — Preliminary  Matter. 

THERE  is  a  class  of  members  in  structures  which  is  subjected 
to  compressive  stress,  and  yet  those  members  do  not  fail  en- 
tirely by  compression.  The  axes  of  these  pieces  coincide,  as 
nearly  as  possible,  with  the  line  of  action  of  the  resultant  of 
the  external  forces,  yet  their  lengths  are  so  great,  compared 
with  their  lateral  dimensions,  that  they  deflect  laterally,  and 
failure  finally  takes  place  by  combined  compression  and  bend- 
ing. Such  pieces  are  called  "  long  columns,"  and  the  applica- 
tion to  them,  of  the  common  theory  of  flexure,  has  been  made 
in  Art.  25. 

Two  different  formulae  have  been  established  for  use  in 
estimating  the  resistance  of  long  columns ;  they  are  known  as 
"  Gordon's  Formula  "  and  "  Hodgkinson's  Formula."  Neither 
Gordon  nor  Hodgkinson,  however,  gave  the  original  demon- 
stration of  either  formula. 

The  form  known  as  Gordon's  formula  was  originally  dem- 
onstrated and  established  by  Thomas  Tredgold  ("  Strength  of 
Cast  Iron  and  other  Metals,"  etc.),  for  rectangular  and  round 
columns,  while  that  known  as  Hodgkinson's  formula  (demon- 
strated in  Art.  25)  was  first  given  by  Euler. 

In  1840,  however,  Eaton  Hodgkinson,  F.R.S.,  published  the 
results  of  some  most  valuable  experiments  made  by  himself,  in 
cast  and  wrought  iron  columns  (Experimental  Researches  on 
the  Strength  of  Pillars  of  Cast  Iron  and  other  Materials;  Phil. 


410 


LONG   COLUMNS. 


[Art.  49. 


Trans,  of  the  Royal  Society,  Part  II.,  1840),  and  from  these 
experiments  he  determined  empirical  coefficients  applicable  to 
Ruler's  formula,  on  which  account  it  has  since  been  called 
Hodgkinson's  formula. 

Mr.  Lewis  Gordon  deduced  from  the  same  experiments 
some  empirical  coefficients  for  Tredgold's  formula,  since  which 
time,  Gordon's  formula  has  been  known. 

The  latter  is  now  in  almost,  if  not  quite,  universal  use 
among  engineers,  and  will  be  completely  given  in  the  next  Ar- 
ticle. Hodgkinson's  coefficients  and  formula  will  be  given 
farther  on. 

Before  taking  up  either,  however,  it  will  be  useful  and  con- 
venient to  determine  the  moments  of  inertia  and  squares  of 
the  radii  of  gyration  of  the  various  forms  of  cross  sections  of 
the  columns  now  in  common  use. 

It  will  also  be  both  convenient 
and  important  to  determine  the  con- 
ditions which  exist  with  an  isotropic 
character  of  section  in  respect  to 
the  moment  of  inertia. 

In  Fig.  la  let  BC  be  any  figure 
whose  area  is  A,  and  and  whose 
centre  of  gravity  is  at  O.  In  the 
plane  of  that  figure  let  any  arbitrary 
system  of  rectangular  co-ordinates 


Fig.  l.a 


X',  Y  be  chosen  and  let  XY  be  any  other  system  having  the 
same  origin  ;  also,  let  x,  y  and  x,  y  be  the  co-ordinates  of  the 
element  D  of  the  surface  A,  in  the  two  systems.  There  will 
then  result  : 

x  =  x  cos  a  -j-  y  sin  a. 


y    COS 


—  X    SIH  Cf. 


The  moments  of  inertia  of  the  surface  about  the  axes  y  and  x 
will  then  be  : 


Art.  49.]  MOMENTS   OF  INERTIA.  41 1 

\x2dA  =  cos2  a\x'2dA  -f  2  sin  a  cos  cAx'y'dA  -f  sin*  ot\y'2dA. 

\y*dA  =  cos2  a\y'2dA  —  2  sin  a  cos  a\x'y'dA  -\-  sin2  a\x'2dA. 

If  x  and  y  are  to  be  so  chosen  that  they  are  principal  axes, 
then  must  \xydA  —  o,  or : 

O  —   \xy dA  =  sin  a  cos  a\y'*dA  -\-  (cos2  a  —  sin2  a)  \x'y'dA 

i 

—  sin  a  cos  a\x'2dA (l#) 

[x'y'dA 


2 

tan  2  a  = 


\x'2dA  -     (y'2dA 


Hence,  since  tan  2ot  =  tan  (180  -f-  2^),  there  will   always  be 
two  principal  axes  90°  apart. 

Now,  if  \ x'y'dA  =  o,  while  no  other  condition  is  imposed, 

tan  2a  =  o.     This  makes  a  =  o  or  90°  ;  i.e.,  X'  Y'  are  the  prin- 
cipal axes. 

If,  however,  \ x'y'dA  =  o,  while  a  is  neither  o  nor  90°,  Eq. 
i#   becomes  : 


(y'2dA  --  (x'2dA  =  o; 


or: 


412  MOMENTS  OF  INERTIA.  [Art.  49. 

tan  2a  =  — ,  i.e.,  indeterminate. 

This  shows  that  any  axis  is  a  principal  axis  ;  also,  that : 

\x*dA  —  \fdA  =  \x'*dA  =  [y'2dA. 
J  J  J  J 

Hence  the  surface  is  completely  isotropic  in  reference  to  its 
moment  of  inertia ;  or,  its  moment  of  inertia  is  the  same  about 
every  axis  lying  in  it  and  passing  through  its  centre  of  gravity. 

It  has  been  seen  that  this  condition  exists  where  there  are 
two  different  rectangular  systems,  for  which 

\xydA  —  \xydA  =  o ; 


but  the  first  of  these  holds  true  if  either  x  or  y  is  an  axis  of 
symmetry,  and  the  latter,  if  either  x'  or  y  is  an  axis  of  sym- 
metry. 

Hence,  if  the  surface  has  two  axes  of  symmetry  not  at  right 
angles  to  each  other,  its  moment  of  inertia  is  the  same  about  all 
axes  passing  through  its  centre  of  gravity  and  lying  in  it. 

Eqs.  (itf)  and  the  two  preceding  it  also  show  that  the  same 
condition  obtains,  if  the  moments  of  inertia  about  four  axes  at 
right  angles  to  each  other,  in  pairs,  are  equal. 

In  the  case  of  such  a  surface,  therefore,  it  will  only  be  nec- 
essary to  compute  the  moment  of  inertia  about  such  an  axis 
as  will  make  the  simplest  operation. 

Since  a  column  fails  partly  by  flexure,  it  is  manifest  that  the 
moment  of  inertia  of  its  cross  section  should  be  the  largest  possible 
about  an  axis  passing  through  its  centre  of  gravity,  and  normal 
to  the  plane  of  flexure. 


Art.  49.] 


BOX  COLUMN. 


413 


Box  Column  of  Plates  and  Angles. 


rig.  i  snows  tne  cross  section  01 

T        1 

1  '     I* 

H 

a  box  column  composed  of  4  plates 

1            | 
J 

f  !  1. 

3 

and   4  or  8   equal  legged    Ls-     FB 

i 

<      1 

and    CD  intersect   at  the  centre  of 

FH-  
1 

1 

a 

gravity  of  the  cross  section. 

1 

1                 j  — 

I 
1     i      ri 

-i 

If  the  4  Ls  shown  in  dotted  lines 

.  i       r1 

i 

\ 

are  omitted,  the  moment  of  inertia 

1                             !                     ^        1 

D 

about  FB  will  be  : 

Fig.  1. 

-a)(d- 


a(d—  2s 


6 


JT] 


If  the  dotted  LS  are  not  omitted  : 


r  -        4 
6 


(d 


(2  ^  + 


2 


-[' 


—  a)  (d  —  2aJ  +  a  (d  —  2s)* 
~3 


If  the  4  LS  shown  in  dotted  lines  are  omitted,  the  moment 
of  inertia  about  CD  will  be  : 


__, 
~"" 


^  +  (^L 


-I- 


414 


MOMENTS  OF  INERTIA. 


[Art.  49. 


If  the  dotted  angles  are  not  omitted  : 


,  _  t'fr       a  \(w  +  2t  +  2sJ>  —  (w  — 

J-     —  —    '    ~^~~     "T"  "       ^ 

6  6t 


(s-d)\(w 


6 


(d-2s) 


12 


'  ™ 


If  latticing  is  used  instead  of  the  two  plates  bt] ',  t'  becomes 
equal  to  zero,  and  the  first  term  in  the  second  member  of  each 
of  the  above  equations  disappears. 

If  A  represents  the  area  of  the  cross  section,  and  r  the 
radius  of  gyration: 


(5) 


j, 5 f. 


1 

L- 

e-10- 

^ 

P1 

F     ~r 
1 

> 

{ 

I 

\  — 

Injj 

fit. 


Box  Column  of  Plates  and  Channels. 

Fig.  2  shows  a  normal  cross  sec- 
tion of  this  column.  FB  and  CD 
-B  intersect  in  the  centre  of  gravity  of 
the  cross  section.  As  in  the  pre- 
ceding Fig.,  these  lines  are  lines  of 
symmetry.  The  moment  of  inertia 
about  FB  is : 


•    (6) 


The  moment  of  inertia  about  CD  is : 


_  t'ift       2a  (w  +  2t  4- 
~~~" 


—  20)  (w 


12 


Art.  49.]  COLUMN  OF  PLATES  AND  ANGLES.  41$ 

If  latticing  takes  the  place  of  the  two  plates  bt  ',  all  terms 
in  the  second  members  of  Eqs.  (6)  and  (7)  involving  t'  will  dis- 
appear. The  moment  of  inertia  about  FB  then  becomes  : 


/=   (f  +   /)  rf»  -  S  (d  - 

and  that  about  CD  : 

.         2a  (W  +    2t   +    2S*   +    (       —  20 


12 

(Radius  of  gyratioii)*  =  r*  =  —  ;  in  which  A 

yit 

is  the  area  of  whole  section. 

Eqs.  (7)  and  (9)  may  also    take  the  forms 
given  in  Eqs.  (15)  and  (16). 

. 

Built  Column  of  Plates  and  Angles. 


,   , 


— B 


i  *  41 i 
Fig.  3  shows  a  normal  cross  section  of  this  '« — &•£—-* 

column   with  the  two    axes  of  symmetry,  FB  Fig,  3. 

and  CD,  intersecting  at  its  centre  of  gravity. 
The  moment  of  inertia  about  FB  is  : 


a(d- 


The  moment  of  inertia  about  CD  takes  the  value  : 


_  t'fr       a  (2s  +  ;)3      (s-a)  (2a  +  ty      (d  -  2s)  fl    ,     , 
~6  ~6~  6~  ~^T " '  { 


MOMENTS  OF  INERTIA.     '  [Art.  49. 

If  the  two  plates  bt'  are  omitted,  the  terms  involving  t'  in 
Eqs.  (10)  and  (11)  reduce  to  zero. 

(Radius  of  gyration)*  =  r2  =  —  ;  in  which  A  is  area  of  sec- 
tion. 

i  False  Channel  Section. 


Let  FB  and  CD  intersect  in  the  centre  of 
gravity,  G,  of  the  section.      The  distance  x^  of 
_,G_  ___  iB    G  from  the  back  of  the  channel,  is  : 


.  ---  .    . 

~~ 


I 

};#,  >|  In  which  A  is  area  of  the  cross  section  of 

Fig.  4.  the  channel.     This  is  usually  found  by  taking 

one-tenth  of  the  weight,  in  pounds,  per  yard 
of  the  channel.     Analytically  : 

A  =  2btf  +  t(d-  2/')     .....     (13) 
The  moment  of  inertia  about  CD  then  becomes  : 

2t' 


0 

About  FBy  it  has  the  value  : 


(14) 


(Radius  of  gyration)*  =  r*  —  -j-  . 

The  line  CD  can  be  very  quickly  and  accurately  located  by 
balancing  the  section,  cut  out  of  manilla  paper,  on  a  knife 
edge. 


Art.  49.] 


ANGLE  IRON. 


417 


Eqs.  (7)  and  (9)  may  now  take  the  forms : 


05) 


(.6) 


In  Eqs.  (15)  and  (16)  A  represents  the  area  of  one  channel 
section. 

Angle  Iron  Section.  „     


Fig.  5  represents  this  section 
with  the  two  axes  taken  parallel 
to  the  legs,  passing  through  the 
centre  of  gravity  G.  The  area 
of  cross  section  is  usually  found 
from  the  weight  per  yard.  Ana- 
lytically : 


T" 

.-'  •• 

1 

Q 

| 

i 

i 

,  

R_j 

| 

'  i 

1 

"  ~i  T° 

_J 

|        I                  t              \ 

F 1 


r r — 


A  = 


Again  : 


-t)t  .  .  .  (17) 


A 


The  moment  of  inertia  about  CD  is 


(19) 


t   /' 


(20) 


27 


MOMENTS  OF  INERTIA.  [Art.  49. 

About  FB  : 


If  the  angle  iron  is  equal  legged,  /  becomes  equal  to  /'. 

(Radius  of  gyration}2  =  r2  =  —  . 

A 

As  in  the  case  of  the  C,  xi  and  *'  may  easily  and  ac- 
curately be  found  by  balancing  a  model  of  the  L  section  on  a 
knife  edge. 

Latticed  Column  of  Four  Angles. 

rThe  four  LS  are  held  in  the  relative 
positions  shown  in  Fig.  6  by  latticing, 

F |_B  the  latter  being  riveted  to  the  legs  of 

the    Ls>    Dut    not   shown   in   the    Fig. 
The  Ls  are  equal  legged. 

From   either  Eq.  (20)   or  (21),    the 
moment  of  inertia  of  the  section  of  any 
one  L>  about  an  axis  passing  through 
its  centre  of  gravity  and  parallel  to  <£,  is  : 

/  (/  -  x$  +  /r,3  -(/-/)  (^  -  /)3 
3 

Hence  the  moment  of  inertia  of  the  column  section  of  Fig. 
6,  about  FB,  is  : 


r  =  4/,  +  A          -  x (22) 

A  is  the  area  of  the  column  section,  or  four  times  the  area 
of  one  L  section. 

If  b  is  different  from  3',  the  moment  of  inertia  of  the  col- 
umn section  about  an  axis  passing  through  its  centre  and  par- 
allel to  b1  will  be  found  by  simply  changing  b1  to  b  in  Eq.  (22). 


Art.  49.] 


LATTICED    COLUMNS. 


419 


(Radius  of  gyration)2  —  r2  =  — • . 

A 


Latticed  Columns  of  Plates  and  Angles. 

£  Fig.  £   represents  a  normal   section 

P — I  ,  |^  ] — -[  of  one  of  these  columns.  By  the  aid  of 
Eq.  (22),  the  moment  of  inertia  of  the 
section  about  FB  may  be  written  : 


f1-. 


D 

Fig.  7. 


and  that   about  CD,  remembering  that 
in  /',  b'  is  to  be  changed  to  b : 


(24) 


If  the  plates  are  on  the  sides  parallel  to  b',  then  b  is  to  be 
changed  to  b'  and  b'  to  b  in  Eqs.  (23)  and  (24). 

Fig.  8  represents  the  normal  section  of  the  other  of  these 
columns,  in  which  there  is  no  latticing,  the  column  being  per- 
fectly closed. 

Again,  using  Eq.  (22),  the  moment  of 
inertia  about  the  axis  FB  is : 

r-U 
f 


. . .  (25) 


The  moment  of  inertia  about  CD  is : 


Fig1.  8. 


(26) 


42O 


MOMENTS  OF  INERTIA. 


[Art.  49. 


In  the  /'  in  Eq.  26,  b'  is  to  be  changed  to  b.     Ordinarily, 
b  =  b'  and  /  =  /'. 

(Radius  of  gyration)1  =  r*  =  —  ,  A    being   area   of   cross- 


section. 


Tee  Section. 


The  axis  FB  is  taken  parallel  to  the  head  of  the  tee  section, 

and  CD  perpendicular  to  it, 
while  G  is  its  centre  of  gravity. 
Analytically,  the  area  of  the 
section  is: 


1 

I          1 

^~ 

1 

« 

JT  B 

I 

'    I 

i.   - 

F 

i 

ig 

> 
.9 

.     .     .      (27) 


The  area  may  also  be  taken 
from  the  weight  in  the  usual 


manner. 


.       .       .       (28) 


The  moment  of  inertia  about  FB  is  : 

^+   /   -   X$   -(b- 


.  .  (29) 


The  moment  of  inertia  about  CD  is  : 

tb*       <#'3 


/  = 


12 


(30) 


(Radius  of  gyrationf  =  r2  —  -r  . 

yi 

As  in  the  other  cases,  /^  may  be  located  by  balancing  on  a 
knife  edge. 


Art.  49.] 


FALSE  EYE   SECTION. 


421 


False  Eye  Section. 

If  the  area  is  not  taken  from  the  weight  per  yard,  it  may  be 
written  : 

!  1C 

e 64— 


A 

-      hfl                  (I)                  f'\     /x/                  0/\                                (l~l\ 

1 
1 

-- 

r 

B 

The  moment  of  inertia  about  CD  is  : 

F~~K 

About  / 

12 

1 

i 

"B  it  has  the  value  :                                        Fi  DIQ 

7  = 


(33) 


(Radius  of  gyration)3  =  ra  =  —  . 

A 


Star  Section. 

Fig.   ii  shows  this   section  with  the  different  dimensions. 

The  area  of  cross  section  is : 


A=bt  +  b't1  -  tt'    .     .     .    (34) 
The   moment  of   inertia   about 


f  — 

> 

—  i— 
U' 

u  1— 

.     6_.     --  J 
6-         -*l 

.     . 

1 

I 

1 

-4— 
i 

—  f~ 

^  

j' 
1 

12 


D 

Fig.11 


About  £Z?  the  moment  of  iner- 
tia has  the  value : 


422 


MOMENTS  OF  INERTIA. 


[Art.  49. 


12 


Ordinarily,  /  =  /'. 

(Radius  of  gyration)*  —  r*  —  -3  . 


(36) 


Solid  Rectangular  Section. 


In  Fig.  12  A  —  bh. 

The  moment  of  inertia  about 


-B 


Fig.  12, 


/  ~  —  '  (^ 

"   12    ' 

and  about  CD  : 


12 


(38) 


(Radius  of  gyration)*  =  r2  =  — r  =  —  or  — 
y  ^  ^4        12       12 


If  the  rectangular  section  is  square,  b  =  7z. 

Hollow  Rectangular  Sections. 

The  area  of  the  section  shown  in  Fig.  13  is:  A  —  bh  —  b'ti. 
The    moment     of     inertia 
about  FB  is : 


bh*  -  b'/i* 


,     N 

;  -   •    (39) 


and  that  about  CD  is  : 


c 

\ 

— 

1 
X 

j 

1      ty 

6            > 

Fig.  13, 


Art.  49.] 


CIRCULAR  SECTIONS. 


423 


12 


(40) 


(Radius  of  gyration)*  =  r2  =  — . 

All  the  equations  of  this  case  (except  Eq.  (40)),  just  as  they 
stand,  apply  directly  to  the  rect- 
angular cellular  section  of  Fig.  14, 
considered  in  reference  to  the  axis 
FB.     If  there  were  n  cells  instead  - — 
of  3,  the  space  between  any  adja- 
cent two  would   have   the  width 
V  Fig.14 


<-£ 


Solid  and  Hollow  Circular  Sections. 

First  consider  a  solid  cylindrical  column  whose  cross  section 
has  the  radius  r2,  as  shown  in  Fig.  15. 
The  moment  of  inertia  about  any  di- 
ameter is : 


(Radius  of  fry  rat  ion}3  =  — — 

4/T7Y8 


'2       -i 

Fig.  15.  4 

Next  consider  a  hollow  circular  column  whose  interior  and 
exterior  radii  are  rl  and  r2  respectively.  The  moment  of  iner- 
tia about  any  diameter  is: 

/  =  *W  "  r*)  _      (r,  +M.  (A  =  area)    m    m    (42) 


(Radius  of  gyration)*  •=. 


r*  +  r;  _ 
4 


=  r3. 


MOMENTS  OF  INERTIA. 


[Art.  49. 


As  tables  of  circular  areas  are  very  accessible,  it  may  be 
convenient  to  write : 


12.566' 


or  r2 


12.566 


Phcenix  Section. 

Fig.  16  shows  the  section  of  a  4  segment  Phoenix  column. 

Let  CD  represent  any  axis 
taken  through  the  centre  of  the 
column.  The  moments  of  iner- 
tia of  the  rectangles  bl  about 
axes  through  their  centres  of 
gravity  and  parallel  to  CD  will 
be  very  small  indeed  compared 
with  the  moment  of  inertia  of 
the  whole  section.  The  mo- 
ment of  inertia  of  any  one 
of  these  rectangles,  therefore, 
about  CD,  will  be  taken  as 
equal  to  the  product  of  its  area 
by  the  square  of  the  normal 
distance  from  its  centre  of  gravity  to  the  axis  CD.  The  mo- 
ment of  inertia  of  the  section  about  CD  will  then  be  : 


+ 
2 


/= 


(43) 


The  moment  of  inertia  is  thus  seen  to  be  the  same  about 
all  axes,  a  result  of  the  general  principle  established  in  the  first 
part  of  this  Article. 

The  area  of  the  cross  section  is : 


Art.  49.] 


TRUE  EYE   SECTION. 


425 


A  = 


(Radius  of  gyration)*  =  r*  =  —j  . 


(43*) 


The  moments  of  inertia  of  six  and  eight  segment  columns 
may  be  found  in  precisely  the  same  manner.  The  moments  of 
inertia  of  the  rectangular  sections  of  the  flanges  about  axes 
passing  through  their  centres  of  gravity,  being  veiy  small  indeed 
when  compared  with  the  moment  of  inertia  of  the  whole  sec- 
tion, may  be  neglected  without  sensible  error. 


Let  r  = 


2S 

b~^ 


True  Eye  Section. 
r  is  then  the 


batter,  or  slope,  of  the  under  side 
of  each  flange  to  the  top  or  bot- 
tom of  the  beam  ;  it  ranges  from 
about  one-third  to  essentially  noth- 
ing. 

If  the  area  of  the  cross  section 
is  not  deduced  from  the  weight  : 


Area  of  section 
=  A  =  2bt  +  ht,  +  s(t>  - 


.  (44) 


The  moment  of  inertia  about  ' 
CD  is: 

_  2tfr  +  7/^,3       r(b*  -  tfy 

/r     -IT        -*r 


Fig.17 


(45) 


If  /x  is  very  small  as  compared  with  b,  remembering  that 
—  r  is  then  essentially  equal  to  s,  there  will  result : 


426 


MOMENTS  OF  INERTIA. 


[Art.  49. 


12 


.    .     (46) 


This  formula  is    sufficiently  accurate    for  all  wrought-iron 
and  steel  beams. 

The  amount  of  inertia  about  FB  is  : 


/  = 


-  —  (7/4  -  k 
47-    v 


12 


In  any  of  these  three  cases  : 


(47) 


(Radius  of  gyration)*  •=.  ~— 

A 


(48) 


True  Channel  Section. 


In  Fig.  1 8  let  r  ~  j-—    -  ;  as  before, 
u  —  t  j 

— •$ 1        it  is  the  batter  or  slope  of  the  under  side 

i, 


— B  duced  from  the  weight : 


of  the  flange. 

If  the  area  of  the  section  is  not  de- 


Area  of  section 


s(b  -  /,)  .  .  .  .  (49) 


The  centre  of  gravity,  G,  can  be  found 
by  balancing  a  manilla,  or  other,  pattern 
on  a  knife  edge  ;  or,  analytically  : 


hi*  +%s(6-  /,)  (b  +  2?, 

A 


•    •    (50) 


The  moment  of  inertia  about  CD  is  : 


Art.  49.]  TRUE    CHANNEL   SECTION.  427 


If  /z  is  very  small  compared  with  by  and  remembering  that 
br  is  then  essentially  equal  to  s  ;  this  last  equation  will  become  : 


The  moment  of  inertia  about  /<7?  is  : 


_Ax,.    .    .    .    (52) 


/  =  -        -  .   ----   ,    (53) 


In  any  of  these  three  cases  : 

(Radius  of  gyration}2  =  —    ..........     (54) 

A 


Deck  Section. 

The  head  of  this  section  will  be  considered  circular  in  out- 
line, as  shown  in  Fig.  19.     Let  a  be  the  area  of  the  circle  C. 
If  the  area  of  the  section  is  not  deduced  from  the  weight : 
Area  of  section 

=  A=a  +  (d-  *)/x  +  (b  -  /,)  (/  +  %  s)  .     .     (55) 

If  the   centre  of  gravity,   G,  is  not  found  by  balancing  a 
pattern  on  a  knife  edge,  there  will  result,  analytically  : 

_  a(2d  -  If)  +  tl(d-  Kf  +bf  +  s(6  -  f,)  (t  +•  #  s) 

2A 


428 


MOMENTS  OF  INERTIA. 


[Art.  49. 


JL 


~G 1. 


S     >| 


Fig,19 


The  moment  of  inertia  about  FB  is  : 


(57) 


2S 


in  which  equation  r  =  7 . 

The  moment  of  inertia  about  CD  is : 


Y±  ah2  -f  t*  (d  —  h  —  t  —  s)  -f  tfc  +  -  (6*  —  /,4) 
/=-  -^  --(58) 


If  /.  is  small  as  compared  with  b,  so  that  essentially  —  =  s : 


48 


In  all  cases  : 


Art.  49.]  ANGLE  SECTION.  429 

(Radius  of  gyration)*  =  — r (60) 

A 


Angle  Section  about  Oblique  Axis. 

The  angle  iron  is  here  supposed  to  be  equal  legged,  and  the 
axis  about  which  the  moment  of  inertia  is  taken,  passes  through 
the  centre  of  gravity  (before  found  in  this  Art.)  and  cuts  the 
sides  /  at  an  angle  of  45°.  In  Fig.  20,  G  is  the  centre  of  grav- 
ity and  HK  the  axis. 


The  moment  of  inertia  about  HK  is  : 

2{xf  -  (*,  -  /)*}   +  t\l  - 


3 
If  A  is  the  area  of  cross  section  : 


(Radius  of  gyration)2  =  —  -   ..........     (62) 

A 

If  a  long  column  has  the  same  degree  of  fixedness  or  free- 
dom in  all  directions,  the  least  value  of  the  square  of  the  ra- 
dius of  gyration  must  be  taken  for  insertion  in  Gordon's  for- 
mula, because  in  the  plane  of  that  radius  the  column  will  offer 
the  least  resistance  to  flexure. 


43°  GORDON'S  FORMULA.  [Art.  50. 


Art.  50.  —  Gordon's  Formula  for  Long  Columns. 

Since  flexure  takes  place,  if  a  long  column  is  subjected  to  a 
thrust  in  the  direction  of  its  length,  the  greatest  intensity  of 
stress  in  a  normal  section  of  the  column  may  be  considered  as 
composed  of  two  parts.  In  fact,  the  condition  of  stress  in  any 
normal  section  of  a  long  column  is  that  of  a  uniformly  varying 
system  composed  of  a  uniform  stress  and  a  stress  couple.  In 
order  to  determine  these  two  parts  let  5  represent  the  area  of 
the  normal  cross  section  ;  7,  its  moment  of  inertia  about  an 
axis  normal  to  the  plane  in  which  flexure  takes  place  ;  r,  its 
radius  of  gyration  in  reference  to  the  same  axis  ;  P,  the  magni- 
tude of  the  imposed  thrust  ;  /,  the  greatest  intensity  of  stress 
allowable  in  the  column,  and  A  ,  the  deflection  corresponding 
to/.  Let/'  be  that  part  of  /caused  by  the  direct  effect  of  P, 
and/"  that  part  due  to  flexure  alone.  Then,  if  h  is  the  greatest 
normal  distance  of  any  element  of  the  column  from  the  axis 
about  which  the  moment  of  inertia  is  taken,  by  the  "  common 
theory  of  flexure  :" 


If  the  column  ends  are  round,  c'  —  I  ;  but  if  the  ends  are 
fixed,  the  value  of  c'  will  depend  upon  the  degree  of  fixedness. 
Also, 


Hence, 


Art.  50.]  GORDON'S  FORMULA.  431 

Eq.  (3)  may  be  considered  one  form  of  Gordon's  formula. 
Before  deducing  the  more  common  and  useful  form,  it  will 

/2 

be  necessary  to  show  that  A  =  a  -j-  ;  in  which  expression  a  is 

considered  constant. 

Let  /  be  the  greatest  intensity  of  bending  stress  in  any 
section,  whose  greatest  value  in  the  column  is  /".  By  the 
"  common  theory  "  (taking  the  origin  of  co-ordinates  at  the 
centre  of  gravity  of  the  cross  section  at  one  end  of  the  column, 
and  the  axis  of  x  along  the  centre  line  before  flexure)  : 


dx* "   h  • 

Also, 

Mh                       MJi 
?=—>  and  /  =  ~j-'> (4) 

in  which  equations   E  is   the  coefficient    of  elasticity  M  the 
bending  moment  for  any  section,  and  M0  the  value  of  M  cor- 
responding to  /". 
Hence, 

>=>"£.•- 2=6 1- 

Consequently, 


The  section  located  by  /0  is  that  at  which  the  deflection  is 

greatest,  and  for  which  -~-  =  o.  while  ~=  is   considered   con- 

dx  Eh 

M 
stant.    The  ratio  -r  is  numerical,  though  variable,  being  one  be- 


432  GORDON'S  FORMULA.  [Art.  50. 

tween  quantities  of  the  same  degree.  M0  is  exactly  the  same 
as  Mt  except  that  x,  in  the  latter,  is  displaced  by  /0 ;  there  are 
the  same  number  of  terms  in  each,  and  those  terms  are  multi- 

f/0  f* 
plied  by  the  same  coefficients.     Now  M ' dx*  may  be  so  ar- 

Jo   J/b 

ranged  as  to  have  the  same  number  of  terms  as  jW0,  but  the  co- 
efficients of  those  terms  will  be  different,  and  the  exponents  of  /0 
in  the  former  will  be  greater  by  2  than  the  exponents  of  1Q  in  M^ 
Hence  /02  =  c*l2  (c  being  some  constant)  will  be  a  factor  in  all 
the  terms  of  the  definite  double  integral.  From  these  con- 
siderations it  follows  that, 


Mda? 

=  al>; (6) 


M. 

in  which  a'  is  some  constant.     Consequently, 


It  is  seen  therefore  that  the  quantity  #,  depends  upon  both 
/"  and  E,  and  it  is  ordinarily  considered  constant. 
Since  /  =  Sra,  Eqs.  (i)  and  (7)  give  : 


Eq.  (8)  shows  that  a^  =  a. 
Hence, 


(9) 


Art.  50.] 


UfllVEESITY 


ROUND  ENDS. 


The  integration  by  which  Eq.  (7)  is  obtained, 
between  limits,  causes  everything  to  disappear  which 
depends  upon  the  condition  of  the  ends  of  the  col- 
umn. Consequently  Eq.  (9)  applies  to  all  columns, 
whether  the  ends  are  rounded  or  fixed.  Let  the  lat- 
ter condition  be  assumed,  and  let  it  be  represented  in 
the  adjoining  figure.  Since  the  column  must  be  bent 
symmetrically,  there  must  be  at  least  two  points  of 
contraflexure.  Two  such  points,  only,  may  be  sup- 
posed, since  such  a  supposition  makes  the  distance 
between  any  two  adjacent  points  the  greatest  possible 
and  induces  the  most  unfavorable  condition  of  bend- 
ing for  the  column. 

If  B  and  C  are  the  points  of  contraflexure  sup- 
posed, then  BCw\\\  be  equal  to  a  half  of  AD,  for  each 
half  of  BC  must  be  in  the  same  condition,  so  far  as  flexure  is 
concerned,  as  either  AB  or  CD.  Also,  the  bending  moment  at 
the  section  midway  between  B  and  C  must  be  equal  to  that  at 
A  or  D.  Consequently,  the  free  or  round  end  column  BC 
must  possess  the  same  resistance  as  the  fixed  or  flat  end  col- 
umn AD.  In  Eq.  (9),  therefore,  let  /  =  2BC  -  2/x : 

.,.!..  (10) 


Fig.1 


l±~ 

r* 

Eq.  (10)  is,  consequently,  the  formula  for  free  or  round  end 
columns  with  length  /x. 

The  flat,  or  fixed  end  column  AD,  is  also  of  the  same  re- 
sistance as  the  column  AC,  with  one  end  flat  and  one  end  free 
or  round.  Hence  in  Eq.  (9)  let  there  be  put  /  =  f  AC  =  \l ', 
and  there  will  result,  nearly, 


P  — 


//, 

7* 


2S 


434  GORDON'S  FORMULA.  [Art.  50. 

Eq.  (11)  is,  then,  the  formula  for  a  column  with  one  end 
flat  and  the  other  round.  A  slight  element  of  approximation 
will  ordinarily  enter  Eq.  (n)  on  account  of  the  fact  that  C  is 
not  found  in  the  tangent  at  A  just  as  Eqs.  (9)  and  (10)  are 
based  on  the  supposition  that  A  and  D  lie  exactly  in  the  line 
of  action  of  the  imposed  load. 

If  the  column  is  swelled,  as  shown  in  Fig.  2,  the 
the  moment  of  inertia  /,  and  distance,  h,  become  vari. 

able.     Hence : 

% 


Consequently, 
and, 

5x 

Fig-2 

M 

If,  in  the  reasoning  applied  to  Eq.  (5),  there  be  written  — 

M 
for  My  and  -~  for  Mw  it  will  at  once  be  seen  that  Eq.  (12)  will 

give  precisely  the  same  general  form  of  result  as  Eq.  (5),  but 
the  coefficient  a  will  have  a  different  value.  Farther,  since 
70  -r-  7  can  never  be  less  than  unity,  but  is  in  general  greater,  it 
follows  that,  for  swelled  columns,  a  is  greater  than  for  columns 
that  are  not  swelled.  Although  these  considerations  show  that 
the  value  of  a  will  be  different  in  the  two  classes  of  columns, 
yet  they  also  show  that  the  general  form  for  the  breaking 
weight  Py  whatever  may  be  the  condition  of  the  ends,  will  be 
precisely  the  same  whether  the  columns  are  swelled  or  straight. 
Since  the  swelling  of  a  column  will  give  it  a  greater  resist- 
ance to  bending,/"  will  take  a  correspondingly  less  value,  while 


Art.  50.]  VARIABILITY  OF  "  CONSTANTS."  435 

Pand  5  remain  the  same.  Eq.  (8),  then,  shows  that  if /and  5 
are  unchanged,  P  must  be  increased.  In  other  words,  a  swelled 
column  will  sustain  a  greater  load  than  one  not  swelled  but 
possessing  the  same  kind  and  area  of  cross  section.  This  is 
indeed  true  of  solid  columns,  but  may  not  be,  and  usually  is 
not,  for  reasons  to  be  assigned  hereafter,  true  for  built  columns 
of  shape  iron.  These  reasons  are  not  introduced  in  the  hy- 
pothesis on  which  the  formulae  are  based.- 

Although  the  quantities  f  and  a,  in  Eqs.  (9),  (10)  and  (n), 
are  usually  considered  constant,  they  are  strictly  variable.  Eq. 
(7)  shows  that  a  is  a  function  of  /"  -f-  E.  It  is  by  no  means 
certain  that/"  is  the  same  for  different  forms  of  cross  section, 
or  even  for  different  sections  of  the  same  form,  and  the  very 
variable  character  of  the  coefficient  of  elasticity  is  well  known. 
It  (the  latter)  is  known  not  only  to  vary  with  the  products  of 
different  iron  mills,  but  even  with  the  different  products  of  the 
same  mill. 

Again,  the  greatest  intensity  of  stress,/",  which  can  exist  in 
the  column  varies  not  only  with  different  grades  of  material, 
but  there  is  some  reason  to  believe  that  it  must  also  be  consid- 
ered as  varying  with  the  length  of  the  column.  The  law  gov- 
erning this  last  kind  of  variation,  for  many  sections,  still  needs 
empirical  determination.  It  is  clear,  therefore,  that  both/ and 
a  must  be  considered  empirical  variables. 

The  expense  necessarily  attending  experimental  researches 
on  the  ultimate  resistance  of  long  columns  built  of  American 
material,  has  prevented  the  attainment  of  many  desirable  re- 
sults. Yet  much  very  valuable  work  of  this  kind  has  been 
done. 

In  the  "  Report  of  Progress  of  Work,"  etc.,  made  by  Thomas 
D.  Lovett,  consulting  and  principal  engineer  to  the  trustees  of 
the  Cincinnati  Southern  Railway,  Nov.  I,  1875,  are  found  the 
records  of  some  valuable  experiments  on  wrought-iron  long  col- 
umns. The  results  of  these  experiments  will  be  used  in  fixing 
values  of  a  and/. 


43^  GORDON'S  FORMULA.  [Art.  50. 

If  the  number  of  experiments  were  sufficiently  great,  the 
results  should  be  combined  by  the  "  Method  of  Least  Squares." 
In  the  present  instance,  however,  the  use  of  the  method  is  al- 
together impracticable  in  consequence  of  the  small  number  of 
experiments  of  any  given  class.  It  will  be  seen,  however,  that 
the  combination  of  the  experimental  results  is  not  altogether  of 
a  random  nature. 

Since /and  a  are  to  be  considered  variable  quantities,  let  y 

p 
take  the  place  of  f  and  x  that  of  a  ;  also,  let  /  =  —  represent 

o 

the  mean  intensity  of  stress.     Eq.  (9)  then  takes  the  form : 

y 


in  which  c  *=  I2  4-  r*.  For  round  or  free  end  columns  x  will 
take  the  place  of  4#,  and  of  i.&a  for  columns  with  one  end  round 
and  one  end  flat. 

In  Eq.  (13)  there  are  two  unknown  quantities,^  and  x,  con- 
sequently two  equations  are  required  for  their  determination. 
If  two  columns  of  different  ultimate  resistances  per  unit  of 
section,  and  with  different  values  of  c,  are  broken  in  a  testing 
machine,  and  the  two  sets  of  data  thus  established  separately 
inserted  in  Eq.  (13),  two  equations  will  result  which  will  be  suf- 
ficient to  completely  give  y  and  x.  Those  two  equations  may 
be  written  as  follows  : 

y=p'  (i  -f*'*) (14) 

y=p"(l+c"x} (15) 

The  simple  elimination  of  y  gives: 


Art.  50.]  PIN  ENDS.  437 

Either  Eq.  (14)  or  (15)  will  then  givejj/. 

In  selecting  experimental  results  for  insertion  in  Eq.  (16), 
care  should  be  taken  to  make  the  differences/"  —  /'  and  c  —  c" 
as  large  numerically  as  possible,  in  order  that  the  errors  of  ex- 
periment may  form  the  smallest  possible  proportion  of  the 
first. 

Before  applying  Eq.  (16)  it  would  be  well  to  recognize  the 
condition  of  the  end  of  a  column  resting  on  a  pin,  as  in  pin 
connection  trusses.  The  end  of  a  column  resting  on  a  pin 
might,  at  first  sight,  be  considered  round  or  free  in  a  plane 
normal  to  the  axis  of  the  pin.  The  compressive  strains  exist- 
ing in  the  vicinity  of  the  surface  of  contact  between  the  pin 
and  soffit  of  the  pin  hole,  produce  a  considerable  surface  on 
which  the  frictional  resistance  to  any  relative  movement  is  very 
great.  This  resistance  to  movement  is  not  sufficient  to  pro- 
duce a  "  flat  "  or  "  fixed  "  condition  of  the  column  end,  but 
causes  a  degree  of  constraint  intermediate  between  the  flat  and 
round  condition;  so  that  a  column  with  two  "  pin  ends  "  gives 
an  ultimate  resistance  approximating  to  that  of  a  column  with 
one  round  end  and  one  fixed  end.  The  following  two  cases 
will  then  hereafter  be  recognized  : 

Two  Pin  Ends, 

One  Pin  End  and  one  Flat  End. 

All  the  necessary  data  for  the  treatment  of  the  experiments 
given  in  the  report  of  Mr.  Lovett,  are  found  in  the  following 
table.  The  column  "  Area  "  gives  the  areas  of  normal  cross  sec- 
tions in  square  inches.  The  column  r2  gives  the  squares  of  the 
radii  of  gyration,  in  inches,  about  axes  normal  to  the  plane  of 
bending.  It  is  inferred  from  the  table  and  the  report  under 
consideration  that  the  radii  of  gyration  for  swelled  columns 
belong  to  sections  at  middle  of  columns.  The  c  column  con- 
tains the  squares  of  /  divided  by  r,  both  being  taken  in  the 
same  unit ;  it  is  a  matter  of  indifference  what  that  unit  may  be. 


438 


GORDON'S  FORMULA. 


[Art.  50. 


The  quantities  x,  y  and/  are  found  by  the  formula  (16),  (15)  or 
(14)  and  (13).  The  column  headed  Exp.  contains  the  ultimate 
resistances  in  pounds  per  square  inch,  determined  by  experiment. 


Keystone 


Closed 


Open 


Square 


Phoenix 


An  "  open  "  column  is  one  in  which  the  flanges  of  the  seg- 
ments that  compose  it  are  separated  by  an  open  space;  a 
closed  column  is  one  in  which  no  such  spaces  are  found.  All 
the  columns  treated  in  the  table  are  closed  except  Nos.  2,  3,  4, 
8,25,31,  24,  26,  30,  and  5. 

The  columns  13  and  19  failed  about  axes  giving  ti\t  greatest 
moments  of  inertia  or  radii  of  gyration.  This  was  probably 
due  to  some  cause  equivalent  to  a  less  degree  of  constraint  at 
the  ends  than  was  intended.  For  this  reason  those  two  results 
are  not  used  in  determining  x  and  y.  They  will  be  Noticed 
again. 

An  examination  of  the  table  shows  that  the  flat  end  swelled 
and  open  straight  Keystone  columns  give  about  the  same  ulti- 
mate resistance,  by  experiment,  per  square  inch,  so  long  as  c 
remains  the  same,  though  the  straight  columns  give  the  largest 
results  by  a  little.  Hence  p'  was  taken  as  an  arithmetical 
mean  of  the  experimental  results  of  Nos.  4,  25,  31,  24,  26,  and 
30,  and  c'  at  9,208.  In  the  same  manner/"  was  taken  a  mean 
of  the  experimental  results  for  Nos.  3  and  8,  and  c"  at  3,060. 
The  arithmetical  means  mentioned  are/'  =  25,517  pounds,  and 
/"  —  32,850  pounds.  Substitutions  in  Eqs.  (16)  and  (15)  or 
(14),  then  give : 


x  =  0.00005455  andj^  =  '38,300.00  pounds. 


Art.    50.]  EXPERIMENTAL  RESULTS. 


439 


•auoisXajj 


•uumicr) 


•xiuaoifcj  '03  ug  'my 


rt  o      *"o       C  ^ 

E      E       £       E 


£        E 


i  1 

C        °        '5  w        -r 

o      J2        a         .So. 


8888888888888     8888      88888888      8888 

vo;    co  M  o;^  ^-  g  10  5  q^  q\oo     q\      5     88.  ft       >n     o.covo  t^   ^     "?    q\       t^     19000 


N  O 

N          ro 


t^        M 

fO       co 


,cT4     H'       o'      «'o"     «n        <S 

ICONN  COCOON  CO 


8Q5        Q      Qooop      55        9      Q°° 
oo        g      oooioO      00        o      ooo 

^\O     co  Q        Q  fh  iovo      K.       10    vo  M        ^-vo  •*• 


ro      co  co  ro 


iiS2S53S 


c^  ON  co 


fOH        c> 
Mt^       10 

oi 


•*?•       coco  10  CN  vo   •**•  t^^o   O    co  O»      ^        •^- 
O        OOOOCOTt-Tt-OM^-NOoK       0>  CO 

£*    8  8  8  M*  £  2  S  S  ^  *""•  °"    8*     ^ 


io       0>ONC?,C> 
00*       CO  OO  CX>  oo' 


O        COfO 
t-x          COCOH 

•<*•      t^-  t-^  10 


00      00  1O      CO  00    CO 


-t-O    OfOfOOOvroNfx 
CO    CNOC  OOMMTt-M^OVOO 


?)  O  QioO 
H  >O  t^O^ 
CO  CO  CO*O  CO 


S:    S 


oc      8    88888 


8    888 

8^      ^b  ^b  "V* 
8    04    M 


oo       ro         vo        t^  ^-  O 


44°  GORDON'S  FORMULA.  [Art.  $0. 

y  is  taken  at  the  nearest  hundred.  These  values. of  x  and  j, 
placed  in  Eq.  (13),  give  the  values  in  column  "/." 

Since,  however,  the  resulting  values  of  "/"  were  a  little  too 
large  for  the  swelled  columns,  and  a  little  too  small  for  the 
straight  open  ones,  x  was  allowed  to  remain  as  determined, 
and  y  was  made  36,000.00  for  swelled,  and  39,500.00  for 
straight  open  columns.  The  resulting  values  of  "p  "  are  given 
in  the  table. 

The  differences  between  the  results  in  the  columns  "/" 
and  "  Exp"  are  not  greater  than  experimental  differences. 

Since  x  depends  on  the  condition  of  the  ends  of  the  col- 
umns, as  well  as  on  the  character  of  the  iron,  it  is  reasonable  to 
give  it  the  same  value  for  all  flat  end  Keystone  columns.  Then 
taking  y  at  39,500.00  pounds,  it  will  be  seen  that  Eq.  (13)  gives 
results  agreeing,  as  nearly  as  could  be  expected,  with  those  of 
experiment  for  straight  closed  Keystone  columns  with  flat 
ends. 

No.  5  is  the  only  experiment  with  a  pin  end  Keystone  col- 
umn. As  it  was  also  swelled,  y  has  been  taken  at  36,000.00 
and  x  at  y-ytinr,  so  that  /  would  be  a  little  less  than  the  result 
of  experiment.  As  these  values  depend  on  one  pin  end  ex- 
periment only,  they  should  not  be  considered  very  satisfactory. 
At  the  same  time  corresponding  values  for  other  columns  show 
that  they  cannot  be  very  erroneous. 

Precisely  the  same  general  principles  and  considerations 
governed  the  selection  of  x  and  y  for  the  several  remaining 
classes  of  columns  shown  in  the  table.  The  agreement  be- 
tween the  columns/  and  Exp.  is  as  close  as  could  be  expected. 

The  extraordinary  character  of  Nos.  13  and  19  has  already 
been  noticed.  No.  13  was  intended  to  be  a  pin  end  column, 
but  the  plane  of  flexure  contained  the  axis  of  the  pin.  Now 
if  it  be  considered  a  round  end  column  in  the  plane  of  failure, 
x  will  have  the  value  4  X  iTGoinr  =  TTT<TO>  anc*  the  resulting 
value  of  /  will  be  24,600.00  pounds.  The  result  of  experi- 
ment was  24,000.00  pounds.  Again,  No.  19  was  intended  to 


Art.  5O.]  RESULTING  FORMULA.  441 

be  a  flat  end  column,  but  it  failed  in  the  direction  of  its  great- 
est radius  of  gyration.  Using  the  values  of  x  and  y  for  pin 
ends,  there  will  result  /  =  26,400.00  pounds.  The  result  of 
experiment  was  27,800.00.  The  effect  of  defective  fitting,  etc., 
would  therefore  seem  to  be  the  lessening  of  the  end  constraint 
by  what  may  be  termed  one  degree. 

Expressing  all  the  results  in  concise  formulae,  they  may  be 
written : 

Keystone  Columns. 
Flat  Ends— Swelled  .  . ./  =  -  -36oo°     /2  ;     ...     (17) 


i  - 


18300  r2 


Flat  Ends— j  Open.  )  =         395QQ 


Straight . .   (  Closed  J 


18300 


Pin  Ends—  Swelled  .........  /=-  -j2  ;     .     .     .     (19) 


15000  r* 


Square  Columns. 

Flat  Ends /  =  -  -j,  ;     .     .     .     (20) 

35000  r2 


Pin  Ends =  -_       ^     ^     ^    (2j) 


I       /' 

17000  r3 


19 


442  GORDON'S  FORMULA.  [Art.  5<D. 


Phcenix  Columns. 


Flat  Ends  .................  p  =  -  _  .  (22) 

T  I  __        _ 

50000  r2 

Round  Ends  ...........  .  .  .p  =  -      42°^°    /2  ;     .     .     ..(23) 

12500  r8 

Pin  Ends  (hypothetical)  ____  /  =  -  -y2  ;     .     .     .     (24) 

22700  r* 


American  Bridge  Co.  Columns. 
- 


Flat  Ends  .................  /  =  -      36°°°    /2  ;     .     .     .     (25) 


46000 


Round  Ends  ...............  /  =  -  /2  ;     .     ;    .     (26) 

•r          I         _     _ 

11500  r2 


Pin  Ends  ..................  /  =  -  /2;    .     .    .     (27) 

21500    r2 

The  pin  end  formula  for  the  Phoenix  column  is  based  on 
the  hypothesis  that  the  relation  between  the  values  of  x  for 
flat  and  pin  ends  is  the  same  as  that  existing  in  the  American 
Bridge  Co.  columns,  which  last  is  shown  by  experiment.  This 
is  a  very  unsatisfactory  method,  and  should  not  be  implicitly 
relied  upon. 


Art.  50.]  SWELLED   COLUMNS.  443 

All  values  of  x  for  round  end  columns  are  found  by  multi- 
plying the  corresponding  flat  end  quantities  by  4,  according  to 
Eq.  (10). 

Eqs.  (17)  to  (28),  inclusive,  give  the  ultimate  resistances  of 
the  various  classes  of  columns.  With  great  variations  of  stress 
a  safety  factor  as  high  as  six  or  eight  may  be  used,  or  it  may 
be  as  low  as  three  or  four  if  the  condition  of  stress  is  uniform 
or  essentially  so. 

For  a  complete  account  of  the  details  of  the  foregoing  ex- 
periments, the  original  "  Report  "  must  be  consulted.  The 
consideration  of  the  shades  of  influence  exerted  by  the  differ- 
ent devices  to  produce  a  given  end  condition  have  here  been 
neglected  on  the  ground  that  such  degrees  of  influence  are  too 
small  to  be  involved  in  a  practical  formula. 

Some  important  deductions  bearing  on  built  columns  of  all 
forms  of  cross  section  may  be  drawn  from  the  results  of  these 
experiments.  It  has  already  been  noticed  that  the  swelled 
columns  Nos.  2,  3,  4,  8,  25,  31,  do  not  give  as  great  ultimate  re- 
sistances as  similar  straight  ones;  a  result  perhaps  not  to  be 
expected,  though  the  explanation  is  simple.  Both  internal 
tensile  and  compressive  stresses  are  induced  in  the  originally 
straight  segments  when  they  are  sprung  to  their  proper  curva- 
ture in  the  swelled  column.  Consequently  this  internal  com- 
pressive stress  causes  a  portion  of  the  material  to  reach  its 
ultimate  resistance  much  sooner  than  would  be  the  case  if  the 
columns  were  straight.  Again,  a  slight  increase  of  direct  com- 
pressive stress  is  caused  by  the  inclination  of  the  segments  of 
the  column  to  its  axis.  If  the  segments  could  be  prepared  for 
the  column  without  initial  internal  stress,  the  ultimate  resist- 
ance would  probably  be  considerably  increased. 

A  consideration  of  these  experiments  would  also  seem  to 
indicate  that  a  closed  column  is  somewhat  stronger  than  an 
open  one.  This  is  undoubtedly  due  to  the  fact  that  the  edges 
of  the  segments  are  mutually  supporting  if  they  are  brought  in 
contact  and  held  so  by  complete  closure,  but  not  otherwise. 


444  GORDON'S  FORMULA.  [Art.  50. 

Thus  the  crippling  or  buckling  of  the  individual  parts  of  the 
column  is  delayed,  and  the  ultimate  resistance  increased. 

The  general  principles  which  govern  the  resistance  of  built 
columns  may,  then,  be  summed  up  as  follows: 

TJie  material  should  be  disposed  as  far  as  possible  from  the 
neutral  axis  of  the  cross  section,  thereby  increasing  r  ; 

There  should  be  no  initial  internal  stress  ; 

The  individual  portions  of  the  column  sliould  be  mutually  sup- 
porting ; 

The  individual  portions  of  the  column  should  be  so  firmly 
secured  to  each  other  that  no  relative  motion  can  take  place,  in 
order  that  the  column  may  fail  as  a  whole,  thus  maintaining  the 
original  value  of  r. 

These  considerations,  it  is  to  be  borne  in  mind,  affect  the 
resistance  of  the  column  only ;  it  may  be  advisable  to  sacrifice 
some  elements  of  resistance  in  order  to  attain  accessibility  to 
the  interior  of  the  compression  member,  for  the  purpose  of 
painting.  This  point  may  be  a  very  important  one,  and  should 
never  be  neglected  in  designing  compression  members.  It 
may  be  observed,  however,  that  the  sole  object  is  to  prevent 
oxidation  in  the  interior  of  the  column,  and  if  the  column  is 
perfectly  closed  this  object  is  attained.  Phoenix  columns 
which  have  been  in  the  most  exposed  situations  (in  one  case 
submerged  in  water  at  one  time  for  several  hours)  during 
periods  varying  from  twelve  to  twenty  years,  without  the 
slightest  oxidation  in  the  interior  of  the  columns,  have  come 
within  the  observation  of  the  writer.  Different  results,  how- 
ever, in  other  cases  have  been  found. 

In  the  experiments  detailed  in  Mr.  Lovett's  report  it  is  to 
be  noticed  that  all  deduced  values  of  y  are  less  than  the  ulti- 
mate resistance  of  wrought  iron  in  short  blocks,  and  some, 
though  not  nearly  all,  would  seem  to  indicate  that  this  differ- 
ence increased  slightly  with  the  length  of  the  column.  Fur- 
ther experiments,  therefore,  may  show  that  the  quantity  /has 
some  such  value  as  the  following: 


Art.  50.]  BOUSCAREN'S  EXPERIMENTS.  445 


C  being  a  constant  quantity,  and  fa  function  of  the  reciprocal 
of  the  length. 

In  connection  with  the  experiments  already  detailed,  Mr. 
G.  Bouscaren,  C.E.,  has  given  an  account,  in  the  Trans,  of  the 
Am.  Soc.  of  Civ.  Engs.  for  Dec.,  1880,  of  other  experiments, 
the  results  of  v/hich  are  given  in  the  table  below. 

Column  No.  33  was  composed  of  four  angle  irons, 


4"  x  2y4"  x  Tv, 

^-fp^ 
I* 


arranged  as  shown  in  the  figure.     It  was  swelled 

from  8%;"  X  8%:"  at  the  ends  to   10"  X   10"  at      I  > 


the   centre.     There   was   only  one   experiment 

with  this  form  of  column,  consequently  the  val- 

ues  of  x  and  y  in   Eq.  (13)  could  not  be  deter- 

mined.     The  angle  irons,  however,  were  of  the 

same  manufacture  as  the  iron  of  which  the  Am.  Br.  Co.'s  col- 

umns were  built.     As   a  mere  matter   of  trial,  therefore,  y  is 

taken  at  36,000.00  pounds,  and  x  is  then  found  to  be  -  . 

43000 

This  result  seems  to  indicate  considerable  advantage  in  such 
a  form  of  column,  but  one  experiment  alone  furnishes  insuf- 
cient  basis  for  such  a  deduction. 

The  columns  35  and  36  illustrate  the  effect  of  repeated  stress. 

The  columns  37  to  43,  inclusive,  were  intended  to  furnish 
information  in  regard  to  the  distance  between  the  rivets  in  the 
zigzag  bracing  and  the  thickness  of  the  metal,  in  order  that 
the  column  may  fail  as  a  whole  and  not  by  "  local  buckling." 

Columns  39  and  40  were  each  composed  of  a  single  short 
piece  of  channel  bar  ;  the  others  were  composed  of  two  chan- 
nel bars  held  together  by  zigzag  bracing. 


446 


GORDON'S  FORMULA. 


[Art.  50. 


NO. 

LENGTH. 

AREA. 

r2. 

c  . 

X. 

y- 

P 

Exp. 

ra" 

33 

28'  6" 

5-68 

20.07 

5.828 

43000 

36,000 

31,700 

3i,7=o 

Pin  Ends. 

35 

34'  o" 

7.48 

8  73 





— 

20,053 

4i> 

36 
43 

34'  o" 
26'  7" 

7.48 
0.50 

8.73 
5-95 

— 



— 

— 

23,128 
18,000 

u 

37 

2/6" 

12.08 

19.98 





— 

— 

29,600 

Flat  Ends. 

38 

23'  oo" 

13  48 

?o.6g 





— 

— 

32,300 

* 

39 

24"  1 

6.6 

0.7 





— 

— 

35,4oo 

" 

40 

,  I?,"1  * 

6.6 

0.7 





— 

— 

35i7°° 

" 

4i 

27'  6" 

13-74 

20  79 



—  — 

—  — 

— 

32.400 

11 

42 

27'  6" 

11.05 

21.26 





— 

— 

32,300 

8" 


~    10*" . 


The  following  forms  of  cross  section,  and  observations,  are 
taken  from  Mr.  Bouscaren's  account : 

Wo.  35. — Gave  way  by  pin  crushing  and  splitting 
web  of  channel.  Column  not  injured  otherwise. 

No.  36.  —  Column  No.  35  tested  again  after 
crushed  ends  had  been  cut  off  and  thickening  plates 
riveted  on  with  pin  holes  34  feet  from  centre  to 
centre.  Column  failed  by  deflection. 

No.  43. — Failed  by  bending  sideways  at    right    }  ^ ?_, 

angles  to  pins,  without  buckling  of  metal.    Bracing  Jgj 

\y^"  X  y^' ;  rivets  20"  apart  in  same  flange  and  10"    j  i     x*"     ! 

in  opposite  flanges. 

Wo.  37. — Webs  buckled  in  both  directions, 
^  in  middle  and  one  end  of  column  ;  column  did 
not  bend.     Bracing  2"  x  TV ;  rivets  24"  apart 
in  same  flange  and  12"  in  op- 
posite flanges. 

No.  38. — Failed  in  same  manner  as  No. 
37  and  by  deflection,  simultaneously.  Bracing 
and  rivets  same  as  in  No.  37. 


<f32S 


u 


. 


|   [«- uZ. ^ 

A"!  «*! 


JJf 


No.  39.  —  Failed  by  buckling  of  web  and 
flanges. 

No.  40.  —  Same  as  No.  39.  Failed  by 
buckling  of  web  and  flanges. 


Art.  50.]  BOUSCAREN'S  CONCLUSIONS.  447 


No.  41.  —  Column  same  as  No.  37  with  riv- 


ets  spaced  20",  in  same  flange,  instead  of  24".      ! 

Failed   by  buckling  of  web  and  bending  in      j  ; 

both  directions,  simultaneously.  IP 

_  A^.  42.  —  Failed  by  buckling  in  plane  of  lat- 

|  ,  —  -^  —  5j-~^  ticing,  without  buckling  of  metal. 

i'j  «"  From    these    experiments    Mr.    Bouscaren 

!  L   x—  -  LJ    concluded  that,  for  the  ratio  of  length  to  diam- 

?   eter  used,  "  the  thickness  of  metal  should  not 

be  less  than  —  of  the  distance  between  supports  transversely, 

.  .  .  .  and  that  the  distance  between  rivets  longitudinally 
should  be  such  that  the  length  of  channel  spanning  it,  con- 
sidered as  a  column,  ....  shall  give  the  same  resistance 
per  square  inch  of  area  as  the  column  itself,  treated  in  the 
same  manner  with  the  same  constant  V"","  (y). 

These  conclusions  are  agreeable  to  that  reached  by  Mr.  B. 
B.  Stoney  :  "  When  the  length  of  a  rectangular  wrought-iron 
tubular  column  does  not  exceed  30  times  its  least  breadth,  it 
fails  by  the  bulging  or  buckling  of  a  short  portion  of  the  plates, 
not  by  the  flexure  of  the  pillar  as  a  whole."  (Theory  of  Strains, 
2d  Edit.,  Art.  334.) 

It  should  be  stated  that  the  experiments  whose  results  have 
been  given  were  made  in  hydraulic  machines  in  which  the 
forces  were  not  weighed,  consequently  the  results  involve  the 
"packing"  friction,  which  was  probably  not  great,  however. 

In  applying  Eqs.  (9),  (10),  and  (u)  to  solid  cast-iron  col- 
umns, there  may  be  taken,  approximately: 


f  =  80000.00  pounds,  and  a  = 

For  solid  wrought-iron  columns,  approximately  : 
/  =  36000.00  pounds,  and  a  = 


448  GORDON  S  FORMULA.  [Art.  50. 

Experiments  on  steel  columns  are  still  lacking.  Mr.  B. 
Baker,  in  his  "  Beams,  Columns,  and  Arches,"  gives  for 

Mild  Steel,     f  —    67000.00  pounds,  and  a  = 
Strong  Steel,  f  =  1 14000.00  pounds,  and  a  = 

These,  however,  must  be  considered  only  loose  approxima- 
tions for  the  ultimate  resistance. 

In  the  "  Trans,  of  Am.  Soc.  Civ.  Engrs.,"  for  Oct.  1880,  are 
given  the  .following  formulae  for  ultimate  resistance  of  wrought- 
iron  columns,  designed  several  years  since  by  C.  Shaler  Smith, 
C.E.: 

Square  Column. 

FLAT   ENDS.  ONE   PIN  END.  TWO   PIN  ENDS. 

_  38500  =  38500 

I  /Q 


5820  d2  3000  d*  1900  d' 


Phoenix  Column. 

42500         _    40000  36600 

=  i  +  _i_i!'        i  ,   i~f* '  ii   * ^ 

^4500^          ^225oa?3  "1500^ 


American  Br.  Co.  Column. 


36500  36500            36500 

—  .  T)    = — .  fl    =Z   — 

T         /3  Jr                         r  •/•*'•-.                         T        /2 

\—L  j  _J L_  L  i-| L_  _L 

3750  d2  22$Qd2                                   1750  ^a 


Art.  51.]  COMMON  AND  PHCENIX   COLUMNS.  449 


Common  Column. 


Lattice 


36500          36500      .     36500 
f  =  t    i  /••   '"  ,,_!__?•   f=  1  +  _L_£' 
2700  d2  i$oo  d2  1200  d?2 

The  formula  for  "  square  columns  "  may  be  used,  without 
much  error,  for  the  common  chord  section  composed  of  two 
channel  bars  and  plates,  with  the  axis  of  the  pin  passing 
through  the  centre  of  gravity  of  the  cross  section. 

Compression  members  composed  of  two  channels  connected 
by  zigzag  bracing,  may  be  treated  by  the  same  formula  after 
putting  36,000.00  for  39,000.00  in  Eqs.  (21)  and  (22). 


Art.  51. — Experiments  on  Phoenix  Columns,*  Latticed  Channel  Columns 

and  Channels. 

In  May  and  July,  1873,  some  experiments  were  made  at 
Phcenixville,  Penn.,  on  full  sized  Phcenix  columns,  by  the 
Phoenix  Iron  Co.  The  results  of  these  experiments  are  given 
in  column  headed  "  Experiment"  while  the  column  headed 
"/"  contains  the  results  of  the  application  of  the  formula 
established  in  the  preceding  Article : 


42000 

42000           ,  , 

I  1 

ii  4  ;" 

50000  r2 

1  50000  r2 

*  The  preceding  Article  was  written  as  a  lecture  and  read  to  the  Class  in  Civil 
Engineering  at  the  Rensselaer  Polytechnic  Institute  nearly  a  year  before  this  Article 
was  written  ;  it  has,  therefore,  been  allowed  to  stand  without  change. 
29 


450 


PHCENIX  EXPERIMENTS. 


[Art.  51. 


according  as  the  ends  are  "  flat "  or  "  round."     All  columns  are 
"4  segment "  ones. 


TABLE  I. 


DATE. 

ENDS. 

AREA. 

LENGTH. 

r2. 

/2-i-r2. 

Experiment. 

/. 

May  3,  1873.  .  . 

Flat  .  .  . 

Sq.  Ins. 

5.84 

Feet. 

23.81 

4.IO 

I9.950 

Lbs. 
30,274.00 

Lbs. 
30,000.00 

May  3,  1873... 

Round.  . 

5-95 

24.00 

4.IO 

20,230 

16,387.00 

16,040.00 

May  3,  1873.  .  . 

Flat  .  .  . 

10.21 

23.38 

8.68 

9,065 

36,419.00 

35,600.00 

Mays,  1873... 

Flat  .  .  . 

8.50 

22.71 

8.00 

9,282 

38,235.00 

35,430.00 

July  19,  1873.. 

Flat  .  .  . 

13.31 

23.20 

8-47 

Q^SI 

32,742.00 

35,500.00 

July  19,  1873.. 

Flat  .  .  . 

12.85 

23.20 

8-47 

9>T5i 

35,408.00 

35,500.00 

In  applying  the  formula  the  length  was  reduced  to  inches, 
in  order  to  bring  it  to  the  same  unit  as  that  in  which  the  radius 
of  gyration,  rt  is  expressed. 

The  columns  "Experiment"  and  "p"  are  each,  of  course, 
per  square  inch. 

It  is  seen  that  the  experimental  results,  and  those  by  Gor- 
don's formula,  give  a  very  close  and  satisfactory  agreement. 
It  is  also  seen  that  the  analytical  relation  between  flat  and 
round  ends  is  a  true  one. 

The  square  of  the  radius  of  gyration,  4.10,  was  taken  the 
same  for  the  first  and  second  columns  because  their  normal 
sectional  areas  are  so  nearly  the  same.  The  value  4.10  belongs 
to  a  4  segment  column,  whose  area  is  5.88  sq.  ins. 

The  same  observation  applies  to  the  last  two  columns.  The 
value  8.47  belongs  to  a  column  whose  area  of  cross  section  is 
13.08  square  inches. 

A  most  valuable  and  instructive  set  of  experiments  on  Phoe- 
nix columns  was  also  made  in  the  large  testing  machine  at  the 


Art.  51.]      EXPERIMENTS  AT    WATERTOWN,   MASS. 


45  J 


U.  S.  arsenal  at  Watertown,  Mass.,  under  the  direction  of 
Messrs.  Clark,  Reeves  &  Co.,  the  results  of  which  were  pre- 
sented to  the  American  Society  of  Civil  Engineers  at  the  I3th 
annual  convention,  June  15,1881.  The  value  of  these  experi- 
ments is  enhanced  by  the  fact  that  they  were  made  on  full 
sized  columns,  such  in  reality  as  are  used  in  ordinary  bridge 
construction. 

In  the  following  table  are  given  the  results  of  these  experi- 
ments, as  well  as  those  of  several  formulae  presently  to  be  ex- 
plained. 

The  following  is  a  portion  of  the  notation : 

/  =  length  in  inches  ; 

r  =  radius  of  gyration  in  inches  ; 
E.  L.  =  elastic  limit  in  pounds  per  square  inch ; 
Exp.  =  ultimate  resistance  in  pounds  per  square  inch. 

TABLE   II. 


NO. 

LENGTH. 

AREA. 

,-. 

/-*-  r. 

M 

E.L. 

Exp. 

A- 

e. 

,: 

Feet. 

Sq.  in. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

28 

2.062 

8.94 

112 

12,544 



35,15° 

32,550 

34,488 



2 

28 

2.181 

8-94 

112 

J2,544 



34,  '5° 

32,55° 

34,488 

•— 

3 

25 

2-233 

8.94 

IOO 

10,000 

27,960 

35,27° 

34,000 

35,°4° 



4 

25 

2.100 

8.94 

100 

10,000 



35,°4° 

34,000 

35,040 



5 

22 

2-371 

8.94 

88 

7,744 



35,57° 

35,42° 

35,592 



6 

7 

22 
19 

2-3" 

2.023 

8.94 
8-94 

88 
75 

7,744 



34,36° 
35,365 

35,42o 
36,800 

35,592 
36,144 



8 

If 

2.087 

8.94 

76 

5,776 

29,290 

36,901 

36,800 



9 

16 

2.OOO 

8.94 

64 

4,096 



36,58° 

38,130 

36,606 



10 

16 

2.000 

8-94 

64 

4,056 



36,580 

38,130 

36,696 

- 

it 

13 

2.185 

8.94 

'52 

2,704 

28,890 

36,857 

39,40° 

37,248 



12 

13 

2.069 

8-94 

52 

2,704 

. 

37,200 

39,400 

37,248 

> 

13 

10 

2-248 

f.94 

40 

i,  600 

26.940 

36,480 

40,700 

37,800 



H 

IO 

2-339 

8.9J 

43 

1,600 

28,360. 

36,397 

40,700 

37,800 



15 

7 

2.265 

8-94 

28 

784 

29,350 

42,200 

38,352 

40,360 

16 

7 

1.962 

8.94 

28 

784. 

29,59° 

43,3°° 

42,200 

38,352 

40.360 

17 

4 

2.031 

8-94 

16 

256 

49,5°o 

44,77° 

46,300 

18 

4 

2.II9 

8.94 

16 

256 

28,050 

51,240 

44,77° 



46.300 

19 

8  ins. 

11.903 

8.04 

2.7 

7.29 



57,130 

69,600 



57,14° 

£O 

8  ins. 

1  !  .903 

8.94 

2.7 

7.29 

. 

57,300 

69,600 

. 

57.14° 

21 

=5'  2.65" 

18.300 

19-37 

68.8 



36,010 

37,600 

36,666 

22 

8'  9" 

18.300 

'9-37 

24 

576 

29,5'° 

42,1SO 

42,840 

42,160 

In  determining  r*  for  Nos.  I  to  20,  inclusive,  a  column  whose 


452  NEW  FORMULA.  [Art.  51. 

area  of  cross  section  was  12.23  square  inches  was  taken.  The 
areas  of  the  actual  cross  sections  varied  so  little  from  this  quan- 
tity, that  the  resulting  value  of  r2  was  assumed  to  belong  to  all 
of  the  first  20  columns.  All  the  columns  were  tested  with  flat 
ends. 

An  application  of  Eq.  (i)  to  these  columns  reveals  consid- 
erable discrepancies  between  the  results  of  that  formula  and 
the  quantities  given  in  the  column  "  Exp"  of  the  table,  when 
the  values  of  /  -f-  r  become  comparatively  small,  as  was  antici- 
pated in  the  preceding  article.  Instead  of  the  constant  42,000 
jn  the  numerator  of  Gordon's  formula,  these  experiments  show 
that  a  variable  quantity  must  be  used,  which  shall  increase  as 
/  -T-  r  decreases,  or  as  r  ~  /  increases. 

After  several  trials  it  was  found  that  the  following  modified 
form  of  Gordon's  formula  would  give  tolerable  results  through- 
out the  entire  range  of  the  experiments  : 

/      .    2r\ 
40000  ( i  +  -r] 

'  A  =  -    A — 4Z  .......  (2) 


50000     r2 

The  results  of  Eq.  (2)  are  given  in  the  column  of  the  table 
headed/!.  The  agreement  between  the  two  columns  is  not  as 
close  as  could  be  desired,  yet  the  discrepancies  are  not  suffi- 
ciently great  to  vitiate  the  safe  use  of  the  formula. 

In  the  following  figure,  the  Watertown  experiments,  as  well 
as  those  of  Mr.  Bouscaren  and  the  Phcenix  Iron  Co.  (given  in 
this  and  the  preceding' Article),  are  shown  by  diagram.  The 
different  classes  of  experiments  are  indicated  as  shown.  The 
experimental  curve  is  drawn  with  particular  reference  to  the 
Watertown  experiments,  for  it  is  then  found  to  be  properly 
located  in  reference  to  the  others.  The  other  curve  expresses 
Gordon's  formula  according  to  Eq.  (2).  It  would  not  be  diffi- 
cult to  find  an  equation  which  would  fit  the  experimental 


Art.  5i.] 


GRAPHICAL  REPRESENTA  TION. 


453 


curve  very  closely  throughout  the  range  of  the  experiments, 
but  it  would  not  be  as  simple  as  Eq.  (2),  or  as  two  others  to  be 
shortly  given. 

It  is  interesting  and  important  to  observe  that  each  experi- 
mental value  in  the  diagram  (which  is  a  mean  of  two,  belong- 
ing to  columns  of  the  same  length,  in  the  table),  lies  on  or 
exceedingly  close  to  the  curve,  with  the  exceptions  of  those 
shown  at  a  and  b.  a  corresponds  to  a  mean  of  Nos.  17  and  18, 
and  is  abnormally  high  ;  b  shows  the  mean  of  Nos.  13  and  14, 
and  is  abnormally  low. 


72000 


60000 


140       120       100       80       60 


It  may  be  observed  that  the  experimental  curve  is  nearly  a 
straight  line  from  a  point  just  above  b  to  the  extreme  left  of 
the  diagram.  For  that  portion  of  the  curve,  therefore,  the 
following  formula  applies  very  closely  : 


/  =  39>640  -  46  - 


(3) 


The  results  of  this  formula  are  given  in  the  column  headed 
"/'."     The  table,  in  connection  with  the  diagram,  shows  that 


454  FORMULA   FOR   SHORT  COLUMNS.  [Art.  51 

this  formula  may  be  used  with  accuracy  for  values  of  /  -f-  r 
lying  between  30  and  140,  and  further  experiments  may  pos- 
sibly show  that  it  is  applicable  above  the  latter  limit. 

For  values  of  /  -i-  r  less  than  30,  the  following  formula  will 
be  found  to  give  results  approximating  very  closely  to  the  ex- 
perimental curve  : 

/'  =  64,700  -  4,600  i/I     .....       (4) 


The  results  of  the  application  of  this  formula  are  given  in 
the  column  headed  "/"." 

The  extreme  simplicity  of  Eqs.  (3)  and  (4)  makes  it  a  mat- 
ter of  great  interest  and  importance  to  determine,  by  other 
experiments  covering  extended  ranges  of  /  -f-  r,  whether  those 
forms,  with  different  constants,  may  not  apply  to  shapes  other 
than  that  of  the  Phcenix  column. 

The  inapplicability  of  the  true  long  column  formulae,  when 

-  is  found  below  certain  limits,  which  is  shown  in  Art.  25,  fur- 

nishes a  proper  foundation  for  thoroughly  empirical  formulae, 
such  as  those  expressed  in  Eqs.  (3)  and  (4). 

By  Eq.  (4),  the  ultimate  resistance  of  Phcenix  wrought 
iron  to  pure  compression  would  be  about  60,000  pounds  per 
square  inch. 

The  results  of  the  application  of  Eqs.  (3)  and  (4)  to  Bou- 
scaren's  and  the  Phcenix  experiments  are  not  given,  but  the 
diagram  shows  clearly  that  they  would  be  satisfactory.  Data 
sufficient  for  the  application  are  given  in  this  and  the  preced- 
ing article. 

The  following  is  the  record  of  the  Phcenix  tests  of  the  very 
short  columns  shown  at  c,  d  and  e  in  the  diagram.  It  is  a  ques- 
tion whether  the  degree  of  distortion  which  accompanied  the 
extremely  high  result  of  65,867  pounds  per  square  inch,  was 
not  considerably  greater  than  that  which  would  characterize 


Art.  51.] 


LATTICED   COLUMNS. 


455 


NO. 

I 

/. 

AREA. 

r2. 

/-H  r. 

/2  -+-  r*. 

EXP. 

A- 

/"• 

Ins. 

8 

Sq.  in. 
6.98 

4.II 

3-95 

15-6 

60,573 

5i,5oo 

55,5oo 

2 

8 

6.98 

4.II 

3-95 

I5.6 

60,387 

5i,5oo 

55,5oo 

3 

4 

5.63 

2-37 

2.6 

6.76 

65,867 

55,8oo 

57,300 

4 

4 

5.63 

2.37 

2.6 

6.76 

65,867 

55,8oo 

57,300 

5 

4 

2-93 

2.25 

2.67 

7-13 

56,889 

55,500 

57,200 

6 

4 

2-93 

2.25 

2.67 

7-13 

55,555 

55,5oo 

57,200 

the  condition  of  "  failure  "  in  an  actual  structure.  This  im- 
portant point  cannot  receive  too  much  attention  in  connection 
with  short  column  tests,  where  the  relative  distortion,  in  the 
condition  of  "  failure,"  is  far  greater  than  that  in  long  columns. 


Latticed  Columns  and  Channels. 

During  1880  and  1881  Col.  T.  T.  S.  Laidley,  U.S.A.,  test- 
ed a  large  number  of  long  columns  composed  of  two  chan- 
nel bars  latticed  in  the  ordinary  manner  (Ex.  Doc.  No.  12, 
47th  Cong.  1st  Session).  These  columns  were  furnished  with 
3^-inch  pin  ends,  and  were  tested  at  Watertown,  Mass.,  in  the 
large  government  machine.  The  adjoin- 
ing figure  shows  the  relative  positions  of 
the  channels  and  pin.  6",  8",  10"  and  12" 
ES  were  employed,  and  all  the  columns, 
the  results  of  whose  tests  are  given  in 
Table  III.,  preserved  the  uniform  distance  of  8  inches  between 
the  channels. 

The  radius  of  gyration,  r,  of  the  cross   section,  given  in 
that  table,  is  in  reference  to  the  axis  of  the  pin. 


456 


LATTICED    COLUMNS. 


[Art.  51. 


All  the  posts  were  single  latticed,  and  the  pitch  of  the 
latticing  (the  distance  apart  of  rivets  in  the  same  flange  of  a  C) 
was  1 8  inches  for  the  6  and  8-inch  channels,  and  22  inches  for 
the  10  and  1 2-inch.  2"  x  %"  latticing  was  used  for  the  6-inch 
Cs  ;  2"  x  ys"  for  the  8  and  lo-inch,  and  2%"  x  3/£"  for  the 
12-inch. 

The  area  of  cross  section  for  the  ES  °f  the  same  depth  in 
different  columns  varied  slightly,  consequently  about  an  av- 
erage area  was  taken. 

TABLE   III. 
Pin   Ends.—T,y2"  Pin. 


NO. 

c. 

AREA  OF 
SECTION  IN  SQ. 
INCHES  (2   CS). 

LENGTH    IN 
INCHES. 

RADIUS   OF 
GYRATION,  INS. 

LENGTH   OVER 
RADIUS  J    OR 

/-*-  r.      . 

/• 

Inches. 

Pounds. 

I 

8 

7-65 

1  60 

3-oo 

53-3 

35,025 

2 

10 

9.70 

200 

3.65 

54-8 

33,920 

3 

6 

4-65 

144 

2-35 

61.3 

34,450 

4 

6 

4-65 

150 

2-35 

63.9 

34,130 

5 

8 

7-65 

200 

3-00 

66.7 

33,79° 

6 

10 

9.70 

250 

3.65 

68.5 

33,770 

7 

6 

4.65 

I  SO 

2-35 

76.7 

34,i8o 

8 

8 

7-65 

240 

3.00 

80.0 

32,375 

9 

12 

12.00 

»       360 

4.44 

81.0 

31,475 

10 

IO 

9.70 

3OO 

3-65 

82.2 

33,oi5 

ii 

6 

4-65 

2IO 

2-35 

89-5 

31,935 

12 

8 

7.65 

280 

3-00 

93-3 

31,800 

13 

10 

9.70 

350 

3-65 

95-9 

30,780 

14 

6 

4-65 

240 

2.35 

IO2.2 

30,085 

15  * 

8 

7-65 

32O 

3-oo 

IO6.7 

29,600 

16 

6 

4.65 

27O 

2-35 

II5.0 

30,820 

17 

8 

7  65 

360 

3.00 

120.0 

25,885 

18 

6 

4-65 

300 

2-35 

127.8 

24,355 

J9 

6 

4-65 

330 

2-35 

I4O.6 

21,330 

20 

6 

4-65 

360 

2-35 

153-4 

15,320 

"/  "  is  the  ultimate  resistance  per  square  inch,  in  pounds. 

All    these    columns    failed    as   wholes,    and    each    result    is 


Art.  51.] 


GRAPHICAL  REPRESENTATION. 


457 


a  mean  of  two.  Other 
columns  of  the  same  set,  ~ 
and  tested  at  the  same 
time,  failed  by  buckling 
of  the  channels ;  they 
cannot,  consequently,  be 
classed  among  long  col- 
umns which  are  so  con- 
structed as  to  fail  as 
wholes. 

The  values  of  /  in 
Table  III.  are  shown 
graphically  in  Plate  I. 
The  ratio  /  -~  r  is  laid 
off  along  the  horizontal 
line  and  the  ultimate  in- 
tensity/ on  the  vertical 
line,  as  shown.  The  full 
curved  line  is  then  the 
experimental  curve  and 
possesses  great  value  of 
a  practical  nature. 
Within  the  limits  of  the 
diagram,  when  the  ratio 


is  known,  the  ultimate 
resistance  of  the  column 
per  square  inch  (/)  can 
be  at  once  accurately 
read  from  the  plate 
without  calculation  or 
scale. 

The   following  equa- 
tion : 


458 


LATTICED    COLUMNS. 


[Art.  51. 


39000 


(5) 


30000     r* 


probably  gives  as  accurate  results  as  any  form  of  Gordon's 
formula.  The  dotted  curve  is  constructed  from  it.  Its  re- 
sults are  seen  to  be  only  tolerably  approximate  between  the 

limits  —  =  50  and  135.  It  possesses  little  value  when  com- 
pared with  the  plate. 

Table  IV.  contains  results  for  columns  of  the  same  set 
which  failed  by  buckling  of  the  individual  channels  of  which 
they  were  composed. 

TABLE    IV. 


NO. 

c. 

LENGTH, 
INCHES, 
/. 

RADIUS     OF 
GYRATION"     IN 
INCHES,  T. 

I 
r 

ULT./  IN    LBS.  PER 
SQ.  INCH. 

CONDITION    OF   ENDS. 

Inches. 

I 

6 

120 

2-35 

5I-I 

36,025 

Flat. 

2 

6 

I  2O 

2-35 

5I-I 

33,740 

One  flat  ;   one  pin. 

3 

10 

126 

3-65 

34-5 

35,450 

Pin. 

4 

12 

I  2O 

4-44 

27.0 

34,245 

Pin. 

5 

12 

I  80 

4.44 

40-5 

34,660 

Pin. 

6 

12 

240 

4-44 

54-o 

33,985 

Pin. 

7 

12 

300 

4-44 

67-5 

33,59° 

Pin. 

If  r'  is  the  radius  of  gyration  in  reference  to  an  axis 
through  the  centre  of  gravity  of  a  single  channel  section,  and 
parallel  to  the  web,  the  following  values  will  hold  for  the  pres- 
ent cases  : 


Art.  51.]          SUCKLING  OF  LATTICED   COLUMNS.  459 


6"C 
8"C 

10"  C 

12"  C 

r'  —  0.58  inches. 
r  =  0.48  inches. 
r'  =  0.69  inches. 
r'  =  0.87  inches. 

Although  the  lattice  rivets  were  alternate  in  the  same  chan- 
nel, each  flange  was  unsupported  for  a  distance  equal  to  the 
pitch,  *>.,  1 8"  for  the  6"  and  8"  ES,  and  22"  for  the  10"  and 
12"  ES.  Hence  : 

For    6"  C  ;  18  -f-  r  =  31.0 

For    8"  C;  I8^r';=  37.4 

For  10"  C  ;  22  -^  r'  =  31.9 

For  12"  C  ;  22  ~  r  =  25.3 

Table  IV.  shows  that  the  column  of  10"  Cs  commenced  to 
fail  by  buckling  of  the  £3  when 

/•*-*•  =  34.5, 
and  when 

22  +  r'  =  31.9; 

that  the  column  of  12"  Cs  commenced  to  fail  similarly  when 
the  length  became  so  small  that 

/  -^-  r  =  27.0, 
while 

22  -f-  r'  =  25.3. 

These  results  would  seem  to  show  that  pin  end  columns 
with  single  but  alternate  latticing  will  begin  to  fail  by  buck- 
ling of  the  channels  when  /-r  r,  for  the  column  as  a  whole, 
becomes  so  small  that  it  is  about  equal  to  the  same  ratio  for  a 
single  channel  between  two  adjacent  rivets  in  the  same  flange. 


LATTICED   COLUMNS.  [Art.  51. 

Nos.  I  and  2  of  Table  IV.  show  that  if  the  ends  possess  a 
greater  degree  of  fixedness,  the  value  of  /  ~-  r  is  much  greater 
when  buckling  begins  to  take  place,  but  the  number  of  experi- 
ments is  not  sufficient  to  indicate  the  exact  amount. 

As  would  be  anticipated  under  the  circumstances,  /  main- 
tains about  the  same  value  for  all  the  columns  in  Table  IV. 
Hence  when  /  ~  r  becomes  so  small  that  buckling  takes  place, 
the  ultimate  resistance  of  the  column  is  independent  of  the 
length. 

The  graphical  representations  of  the  results  given  in  this 
Article  show  that  the  curve  of  ultimate  resistances  has  a  very 
sharp  declivity  for  small  values  of  /-r-  r,  but  that  it  becomes 
nearly  straight  and  horizontal  for  larger  values,  and  that  it 
again  increases  in  declivity  with  a  still  father  increase  in  that 
ratio.  These  phenomena  seem  to  be  much  more  pronounced 
in  the  tubular  variety  of  columns.  They  find  a  simple  and 
obvious  explanation  in  the  fact  that  in  columns  of  moderate 
length  the  deflection  at  the  centre  of  the  column  about  keeps 
pace  (in  the  same  direction)  with  the  movement  of  the  centre 
of  pressure  at  the  ends. 

Plate  I.  shows  (what  was  to  be  anticipated)  that  this  effect 
is  also  much  less  pronounced  with  pin  ends  than  with  flat 
ones,  it  being  borne  in  mind  that  the  phenomena  here  consid- 
ered do  not  produce  the  horizontal  straight  line  which  would 
be  seen  if  Plate  I.  included  less  values  of  /  -f-  r  than  50.  The 
latter  represents  the  buckling  of  the  individual  parts  of  the 
column,  and  not  the  failure  of  the  column  as  a  whole. 

A  few  experiments  by  Col,  Laidley  with  columns  of  the 
same  Cs  as  the  above,  but  with  pins  only  three  inches  in  diam- 
eter, gave  uniformly  less  ultimate  resistance  than  those  with 
three  and  a  half  inch  pins.  Although  this  result  was  to  be 
expected,  the  number  of  experiments  was  not  sufficient  to 
justify  any  quantitative  conclusions  ;  it  can  only  be  stated 
that  the  smaller  the  pin  the  less  will  be  the  ultimate  resist- 
ance. 


Art.  51.] 


FLAT  END    CHANNELS. 


461 


TABLE    V. 
Flat  End  EJ. 


NO. 

c. 

AREA   OF 
SECTION  IN  SQ. 
INCHES. 

/. 

r'  '  . 

/ 
f7' 

VLT.    RESIST.,   IN 
l.BS.   PER  SQ.  INCH 
=  /. 

Inches. 

Inches. 

Inches. 

I 

6 

2.33 

6.OO 

0.58 

10.3 

42,293 

2 

6 

2-33 

17.60 

0.58 

30.3 

36,835 

3 

6 

2-33 

23   90 

0.58 

41.  X 

33,910 

4 

6 

2-33 

48.00 

0.58 

82.6 

28,140 

5 

8 

3.80 

8.00 

0.48 

16.6 

43,295 

6 

8 

3.80 

17.90 

0.48 

37-2 

35,280 

7 

8 

3.80 

23-90 

0.48 

49  7 

35,975 

8 

8 

3  -So 

29.90 

0.48 

62.2 

33,4oo 

9 

8 

3.80 

48.00 

0.48 

99.8 

30,620 

10 

10 

4.85 

10.00 

0.69 

14-5 

35,o8o 

ii 

10 

4-35 

17.90 

0.69 

26.0 

33,820 

12 

IO 

4.85 

23.90 

0.69 

34-7 

34,355 

13 

10 

4-85 

29.90 

0.69 

43-4 

34,050 

14 

IO 

4-85 

48.00 

0.69 

69.6 

34,o8o     . 

15 

12 

6.00 

12.00 

0.87 

13-8 

37,240 

16 

12 

6.00 

17.80 

0.87 

20.5 

36,59° 

I? 

12 

6.00 

23.90 

0.87 

27-5 

36,695 

18 

12 

6.00 

29    90 

0.87 

34-4 

35,150 

19 

12 

6.00 

48.00 

0.87 

55-2 

36,040 

Table  V.  contains  the  results  of  Col.  Laidley's  tests  of  por- 
tions of  the  Cs  used  in  the  columns  which  have  just  been 
treated.  These  portions  had  flat  ends. 

The  moment  of  inertia  of  the  section,  from  which  the 
radius  of  gyration  r  was  computed,  was  taken  about  an  axis 
parallel  to  the  web  of  the  channel  and  passing  through  its 
centre  of  gravity. 

Many  of  the  results  are  means  of  two  tests  each. 

The  results  given  in  Table  V.  are  shown  graphically  in 
Plate  II.  The  values  of  the  ratio  /  -^  r  are  laid  off  on  the  hori- 
zontal base  line,  to  the  left  from  O ;  while  the  values  of  /  in 


462 


CHANNELS  AS  COLUMNS. 


[Art.  51. 


uJ 


pounds  per  square 
inch  are  laid  off  ver- 
tically from  O,  as 
shown.  The  full 
curve  then  repre- 
sents with  great  ac- 
curacy the  experi- 
mental results. 

The  dotted  curve 
represents  the  fol- 
lowing form  of  Gor- 
don's formula  for 
the  ultimate  resist- 
ance in  pounds  per 
square  inch  : 

/=    *TV6) 

63000  r'2 
This    formula   is 


for  all  ordinary  pur- 
poses, between  the 
limits 


-H  /  == 


and 


but  does  not  com- 
pare in  value  with 
the  experimental 
(full)  curve. 


Art.  52.]  LONG   COLUMN  FORMULA.  463 


Art.  52. — Euler's  and  Tredgold's  Forms  of  Long  Column  Formulae. 
<»• 

The  form  of  the  general  formula  given  in  the  preceding  Ar- 
ticle, as  will  presently  be  shown,  does  not  seem  to  be  as  well 
adapted  to  the  expression  of  accurate  results  as  that  of  Euler, 
given  in  Art.  25. 

It  has  already  been  observed  that  the  coefficient  a  (Eq.  (9) 

.;/ 

of  Art.   50),  contains  ±-=  as  a  factor,  in  which  /"  is  the  great- 

£L 

est  intensity  of  bending  stress,  i.e.,  a  part  of  the  quantity  "/  " 
which  is  sought.  The  possible  use  of  the  formula  is  based  on 
the  fact  that  E  is  very  large  in  respect  to/". 

The  existence  of/"  in  a  is  due  to  the  redundant  form  of  Eq. 
(8)  of  the  Article  cited. 

Since,  in  that  Article,/'  =  -  and  a  =  aj1  =  ?  •  (see  Eq. 
(7)),  Eq.  (8)  gives: 

„  _       PI*  __  a'c'p"  PI* 
*         a  Sr*  ~      E     S^  ; 


This  is  Euler's  formula  as  given  in  Eq.  (6)  of  Art.  25.  In 
this  equation  b  has  the  analytical  values  4n*E,  n*E  and  2.25  n*E 
for  ends  fixed,  rounded  and  one  fixed  one  rounded,  respect- 
ively, as  shown  in  Art.  25. 

It  would  seem,  therefore,  that,  since  Eq.  (i)  involves  noth- 
ing variable  in  the  second  member  but  r  ~-  /,  it  ought  to  give 
more  accurate  results  than  Tredgold's  form  of  Art.  50. 

It  was  shown,  however,  in  Art.  25  that  the  common  the- 
ory of  flexure  is  analytically  applicable  only  to  fixed  end  col- 
umns of  wrought  iron,  in  which  the  ratio  of  length  over  radius 


4^4  EULERS  FORMULA.  [Art.  52. 

of  gyration  is  somewhat  greater  than  140;  and  to  round 
end  columns  in  which  that  ratio  is  somewhat  greater  than 
70.  Since  the  implicit  assumption  of  an  indefinitely  small  cross 
section  underlies  the  analytical  treatment  of  long  columns,  it  is 
possible  that  the  analytical  coefficients  and  exponent  may  not 
obtain  far  above  the  limits  indicated  in  Art.  25.  Now,  since 
other  conditions  of  ends  will  lie  between  these  limits,  it  is  seen 
that  both  long  column  formulae  are  strictly  inapplicable  to  a 
large  portion  of  the  columns  designed  by  engineers. 

Fortunately,  a  sufficient  number  of  experiments  have  been 
made  with  full  sized  columns  to  show  that  either  form  of  for- 
mula, when  holding  empirical  quantities  properly  determined, 
will  give  excellent  results.  This  has  already  been  shown  for 
Tredgold's  form,  and  it  will  now  be  seen  that  Euler's  form  may 
be  expected  to  give  still  better  results. 

If,  as  is  usual,  r  is  the  radius  of  gyration  and  /  the  length 
(both  in  the  same  unit),  and  if  both  the  coefficient  and  exponent 

of  —  ,  in  Euler's  general  formula,  be  considered  variable,  the 
following  equation  (see  Art.  25),  may  be  written  : 


For  other  values  (r'  and  /')  of  r  and  /,  the  mean  intensity 
becomes  : 


Dividing   Eq.  (3)  by  Eq.   (2),  then  taking  logarithms  and 
solving  for  x  : 


Art.  52.]  RESULTS  FOR  PH(ENIX  COLUMNS. 


465 


Subtracting  Eq.  (3)  from  Eq.  (2)  and  solving  for  j: 


I/  — 


(5) 


These  formulae  will  first  be  applied  to  results  of  the  experi- 
ments made  on  Phoenix  columns  at  Watertown,  Mass.  These 
results  are  contained  in  Table  II.  of  the  preceding  Article,  and 

"/" 
the  columns  -   -  and  "  Exp"  are  reproduced  in  Table  I.  of  this 

Article.  In  the  latter,  however,  the  column  "  Exp"  contains 
the  means  of  the  various  pairs  of  experiments  whose  results 
are  given  in  the  former. 

TABLE   I. 

Phoenix  Columns. 


T 

i*. 

p. 

r 

**. 

A 

112 

34,650 

34,550 

40.0 

36,440 

39,000 

100 

35,150 

35,000 

28.0 

40,700 

40,630 

88 

35,000 

35,530 

16.0 

50,400 

43,400 

76 

36,130 

36,150 

2-7 

57,200 

53,400 

64 

36,580 

36,900 

68.8 

36,000 

36,570 

52 

37,ooo 

37,800 

24.0 

42,200 

41,400 

1 

Now,  let  there  be  taken : 


-  =    28 
r 


30 


466  EULERS  FORMULA.  [Art.  52. 


/  =  34,650. 


Inserting  these  values  in  Eqs.  (4)  and  (5),  there  will  result  : 

x  =  0.117  andj/  =  59,723. 
Then  let  there  be  written  : 


©0.117 
......    (6) 


The  various  values  of  I  -  }  in  Table  I.,  inserted  in  Eq.  (6), 

give  the  results  shown  in  columns  "/  "  of  that  Table.  They 
are  seen  to  be  much  more  satisfactory,  as  a  whole,  than  those 
given  by  any  form  of  Tredgold's  formula  in  the  preceding 
Articles;  although  Eq.  (2)  of  Art.  51  is  a  little  closer  to  the 

experimental  results  for  values  of  —  less  than  24. 

So  much  of  the  curve  represented  by  Eq.  (6)  as  does  not 
coincide  with  the  experimental  curve,  is  shown  by  the  dotted 
line  in  the  Fig.  of  the  preceding  Article. 

That  curve,  together  with  the  results  given  in  Table  I., 
shows  the  close  agreement  of  Eq.  (6)  with  experiment  for  all 

values  of  -r  from  I  to  •  —  . 
/  112 

y 
It  is  interesting  and  important  to  observe  that  when  --  =  I, 

Eq.  (6)  gives  : 

/  =  60,000  ; 

or  about  the  ultimate  compressive  resistance  of  wrought  iron 
in  cubes. 

An  application  of  Eqs.  (4)  and  (5),  in  the  manner  already 


Art.  52.]          RESULTS  FOR  KEYSTONE   COLUMNS. 


467 


shown,  to  the  results  of  Bouscaren's  experiments  on  Keystone 
columns,  given  in  the  large  table  of  Art.  50,  gave  the  following 
results  for  swelled  Keystone  columns  : 


x  =  0.25     and    y  =  78,000;  or: 

H 


=   78,000 


TABLE   II. 
Keystone  Columns. 


(7) 


SWELLED. 

STRAIGHT. 

Exf. 

/• 

C. 

Exp. 

/• 

326 

33,600 

37,800 

8,718 

25,000 

28,000 

2,991 

28,800 

28,700 

9,391 

27,500 

27,700 

9,646 

24,100 

24,800 

9,157 

30,000 

27,800 

3,130 

36,900 

28,5OO 

3,519 

30,000 

31,350 

9,189 

21,100 

24,900 

4,136 

32,000 

30,700 

9>T57 

25,400 

24,900 

10,714 

27,8OO 

27,300 

Also,  for  straight  Keystone  columns : 

x  =  0.25     and    y  —  87,000;  or: 


=  87,000 


(8) 


The  results  of  the  application  of  these  formulas,  and  the  ex- 


EULEKS  FORMULA.  [Art.  52. 


perimental  results,  are  given  in  Table  II.     The  lengths  and 
other  data  can  be  found  in  the  table  just  cited. 

By  the  same  operations  with    the   square    column    results 
(Bouscaren's)  of  the  same  table,  there  were  found  : 

x  =  0.5,    and    y  —  303,000;  or: 


=  303,000  (-\ (9) 


The  following  columns,  "  Exp"  and  "/  "  contain  the  ex- 
perimental square  column  results  and  those  computed  from 
Eq.  (9). 

c.  Exj>.  /. 

10,414 30,000 30,000  1 

/      Square 

7V133 33'2°° ••••33>°<X4  Columns. 

9,623 3O,2OO 3O,6OO   ) 

Only  "  flat  end  "  experiments  have  been  treated,  for  the 
others  are  utterly  insufficient  in  number  for  the  determination 
of  the  empirical  quantities. 

In  fact,  with  the  exception  of  the  Watertown  experiments 
on  the  Phcenix  columns,  the  number  of  those  with  "  flat  ends  " 
is  not  sufficiently  great,  nor  the  range  of  /  -5-  r  sufficiently  ex- 
tended, to  establish  reliable  formulae. 

In  all  cases,  however,  it  is  to  be  observed  that  the  formulae 
of  this  Article  give  results  more  nearly  agreeing  with  the  ex- 
perimental ones  than  those  computed  from  any  form  of  Tred- 
gold's  or  Gordon's  formula.  It  would  seem  that  this  form  of 
formula  has  not  heretofore  received  the  attention  to  which  its 
importance  and  value  entitle  it. 

Each  of  the  three  Eqs.  (7),  (8)  and  (9),  become  inapplicable 

when  the  value  of  -  is  such  that  "/  "  approaches  the  ultimate 


Art.  53.]  HODGKINSON'S  FORMULA.  469 

compressive    resistance  per  square  inch    of  wrought  iron  in 
short  blocks. 

These  empirical  results  tend  to  give  experimental  confirma- 
tion to  Eider's  formula,  for  the  exponent  and  coefficient  of  \-j\ 

are  seen  to  increase  very  much  as  the  lowest  value  of  c,  in  the 
different  sets  of  experiments,  increases. 


Art.  53.  —  Hodgkinson's  Formulae. 

The  detailed  account  of  the  experiments  on  which  Eaton 
Hodgkinson  based  his  various  formulae  is  given  in  the  Phil. 
Trans,  of  the  Royal  Society  of  London,  for  1840.  His  cast-iron 
columns  were  small  ones,  the  greatest  length  of  which  was  60.5 
inches.  The  greatest  value  of  the  length  divided  by  the  radius 
of  gyration  was  : 


while  the  least  value  of  the  same  ratio  was  : 

~r  =  4  X  oF  =  3°'2  ( 
The  greatest  diameter  was  about  two  inches. 

Let  d  =  diameter  of  column  in  inches. 
Let  /  =  length  of  column  in  feet. 

Then  for  the  breaking  weight  (P)  of  solid  cylindrical  cast 
iron  columns,  when  expressed  in  pounds,  Hodgkinson's  for 
mulae  take  the  shape  : 


470  HODGKINSON'S  FORMULA.  [Art.  53. 

^3.76 
P  =  33>379  777  '  (for  Bunded  ends)    .     .     .     (i) 

P  -  98,922  —  ;  (for  fixed  ends)     ....     (2) 

For  hollow  cylindrical  columns  of  cast  iron  : 

/>76  _  ^3.76 
P=29,I2O-   —     --  •;  (for  rounded  ends)    .     .     (3) 


P=  99,320  --  -^  -  ;  (for  fixed  ends)    ...     (4) 

In  Eqs.  (3)  and  (4),  D  is  the  greater,  or  exterior,  diameter 
of  the  column,  while  d  is  the  interior  diameter.  It  is  to  be 
observed  that  P  is  the  total  breaking  weight  in  pounds. 

The  longest  wrought-iron  solid  cylindrical  column  tested 
by  Hodgkinson  had  a  length  of  90.75  inches  and  a  diameter  of 
about  i.  02  inches.  Hence  the  greatest  ratio  of  length  over 
radius  of  gyration  was  about  90.75  X  4  =  363. 

His  formulae  for  the  total  breaking  weight  of  solid  cylindri- 
cal wrought-iron  columns,  in  pounds,  are  : 


P  =  95,848  -j^  ;  (for  rounded  ends)     ...     (5) 


//3-55 

P  =  299,617  —  ;  (for  fixed  ends)    ....     (6) 

In  his  experiments  on  square  pillars  of  Dantzic  oak,  the 
greatest  dimensions  were  :  length  =  60.5  inches,  and  side  of 
square  section  =  1.75  inches. 

His  longest  red  deal  pillar  was  58  inches  in  length,  and  the 
cross  sections  were  1x1,1x2  and  i  x  3  ;  all  in  inches. 


Art.  53.]  TIMBER   COLUMNS.  4/1 

Hodgkinson  used  Lamande's  experiments  on  French  oak 
in  establishing  a  formula  for  that  material.  In  those  experi- 
ments, the  longest  pillar  had  a  length  of  76.5  inches  and  a 
normal  section  of  2.13  inches  by  2.13  inches. 

Retaining  the  same  notation,  the  following  are  the  total 
breaking  weights,  in  pounds,  of  solid  square  timber  pillars  with 
flat  ends : 

d4 
Dantzic  oak  (dry)  ;     P  =  24,542  —      .      .     .     (7) 


Red  deal  (dry)  ;          />=  17,511^-      .      .-.     (8) 


French  oak  (dry)  ;      P=  15,455-—      .      .     .     (9) 

In  Eqs.  (7),  (8)  and  (9),  "  d"  is  the  side  of  the  square  sec- 
tion of  the  column  in  inches,  while  /is  the  length  in  feet. 

All  the  preceding  formulae  are  to  be  used  only  in  those 
cases  in  which  the  length  exceeds  30  times  the  diameter  or 
side  of  square,  if  the  ends  are  fixed  ;  or  15  times  the  length,  if 
the  ends  are  rounded.  Between  these  limits  and  a  short  block, 
in  which  the  length  is  4  or  5  times  the  diameter  or  less,  the 
following  formula  is  to  be  used  :  Let  C  be  the  ultimate  com- 
pressive  resistance  of  the  material,  per  unit  of  area,  in  short 
blocks,  and  let  A  be  the  area  of  the  normal  section  of  the  col- 
umn ;  then  Hodgkinson's  formula  for  these  columns  of  inter- 
mediate lengths  is  : 


pf  — 

~ 


The  small  size  of  the  columns  experimented  upon  by 
Hodgkinson  militates  very  strongly  against  the  practical  value 
of  his  formulae,  unless  it  should  be  shown  experimentally  that 


472  HODGKINSON'S  CONCLUSIONS.  [Art.  53. 

the  same  formulae  may  be  equally  applicable  to  large  and  small 
columns. 

With  the  greatest  ratio  of  /  over  r,  the  ratio  of  the  resist- 
ance of  a  fixed  end  pillar  over  that  of  one  of  the  same  length 
and  with  rounded  ends  was  about  3.34.  With  the  lowest  value 
of  /over  r,  the  same  ratio  was  about  1.63.  According  to  Eu- 
ler's  formula,  that  ratio  should  have  been  4.  It  is  seen,  there- 
fore, that  with  these  columns  the  common  theory  of  flexure 
failed  far  above  the  limit  given  in  Art.  25. 

From  his  experiments  Hodgkinson  drew  the  following  con- 
clusions : 

The  strength  of  a  pillar  with  one  end  round  and  the  other 
flat,  is  the  arithmetical  mean  between  that  of  a  pillar  of  the 
same  dimensions  with  both  ends  rounded,  and  with  both  ends 
flat. 

A  long  uniform  pillar,  with  its  ends  firmly  fixed,  whether 
by  disks  or  otherwise,  has  the  same- power  to  resist  breaking  as 
a  pillar  of  the  same  diameter  and  half  the  length,  with  the 
ends  rounded  or  turned  so  that  the  force  would  pass  through 
the  axis. 

Long  uniform  cast-iron  pillars  with  both  ends  round  break 
in  one  place  only — the  middle  ;  those  with  both  ends  flat,  near 
each  end  and  at  the  middle  ;  those  with  one  end  round  and 
one  end  flat,  about  one-third  the  length  from  the  round  end. 

The  resistance  of  solid  pillars  with  round  ends  was  increased 
about  one-seventh  by  increasing  the  diameter  at  the  middle. 
Flat-end  pillars  (solid)  had  their  resistances  increased  very 
slightly  by  the  same  means,  but  hollow  pillars  seemed  to  derive 
no  benefit  at  all  by  enlargement  at  the  middle. 

The  resistance  of  flat-end  pillars  was  increased  slightly  by 
the  application  of  disks  to  their  ends. 

Irregular  and  imperfect  fixedness  of  the  ends  may  cause  a 
loss  of  two-thirds,  or  more,  of  the  resistance  with  ends  per- 
fectly fixed. 

Solid  square  cast-iron  pillars  failed  in  diagonal  planes. 


Art.  54.] 


TUBES  AS  COLUMNS. 


or  THE        'r    > 

UNIVERSITY 

oir 


The  relative  resistances  of  columns  of  the  same  length  and 
area  of  cross  section  were  about  as  follows : 

Long,  solid,  round  pillar loo 

"         "      square  pillar 93 

"         "      triangular  pillar no 


Art.  54. — Graphical  Representation  of  Results  of  Long  Column 
Experiments. 

If  the  values  of  /  over  r  (length  9ver  radius  of  gyration),  for 

TABLE   I. 
Tubes.— Flat  Ends. 


NO. 

LENGTH. 

EXT.   DIA. 

THICKNESS. 

AREA. 

l+r. 

ULT.  RESIST.  PER 
SQUARE   INCH. 

Inches. 

Inches. 

Inch. 

Sq.  ins. 

Pounds. 

I 

1  20 

1-5 

O.IO 

0.44 

240 

14,670 

2 

120 

2.00 

0.10 

0.61 

179 

23,206 

3 

1  20 

2-35 

0.23 

1.50 

1  60 

21,900 

4 

1  20 

2.50 

O.  II 

0.80 

141 

29,800 

5 

120 

3-oo 

0.15 

i-35 

120 

27,670 

6 

60 

1.50 

O.IO 

5.44 

120 

31,180 

7 

90 

3-04 

0.17 

1.41 

90 

29,790 

8 

60 

2.OO 

O.  IO 

0.61 

89 

33,300 

9 

120 

4-05 

0.16 

1.9 

87 

26,960 

10 

60 

2-35 

0.22 

1.47 

80 

29,330 

n 

60 

2-34 

O.2I 

1-37 

80 

30,000 

12 

60 

2.50 

O.II 

0.80 

71 

35,ioo 

13 

89 

4.00 

0.24 

2.87 

67 

26,800 

14 

90 

4-05 

O.  12 

1.61 

65 

33,330 

15 

30 

1.50 

O.IO 

0.44 

60 

34,220 

•  16 

60 

4.00 

O.24 

2-85 

45 

32,200 

17 

30 

2.OO 

O.IO 

0.61 

45 

36,980 

18 

30 

2-35 

0.24 

i.  60 

40 

35,660 

19 

30 

2-34 

O.2I 

1.44 

40 

36,000 

20 

29 

2-37 

0.23 

1-55 

39 

36,910 

21 

29 

2-34 

O.2O 

1.36 

39 

39,570 

22 

30 

2.50 

O.TI 

0.80 

35 

36,49° 

23 

28 

3-00 

0.15 

1.41 

28 

37,39° 

24 

28 

4.00 

0.25 

2.85 

21 

48,200  . 

474 


GRAPHICAL   REPRESENTA  TION. 


[Art.  54. 


a  series  of  columns  which  have  been  tested  to  breaking,  be 
accurately  laid  off  on  a  horizontal  scale,  and  if  the  breaking 
weights  per  square  inch  be  laid  off  with  equal  accuracy  on  a 
vertical  scale,  the  resulting  curve  will  represent  the  resistances 
of  all  columns  for  which  /  over  r  lies  within  the  limits  of  the 
experiments,  with  far  more  accuracy  than  any  simple  and  prac- 
ticable formula  that  can  be  devised.  Such  a  curve  for  the 
Watertown  experiments  on  Phoenix  columns  has  already  been 
incidentally  constructed  in  Art.  51. 

•TABLE  II. 
Solid  Rectangular  Pillars. — Flat  Ends. 


NO. 

LENGTH. 

SECTION. 

AREA. 

/-+-  r. 

ULT.  RESIST.  PER 
SQ.  IN. 

Inches. 

Inches. 

Sq.  Ins. 

Pounds. 

I 

I2O 

2.98  x  0.5 

i/5 

822 

8,160 

2 

90 

2.98  X  0.5 

i-5 

643 

2,410 

3 

120 

3.01  x  0.77 

2.31 

540 

3,380 

4 

1  2O 

3  .  oo  x  I  .  oo  • 

3.00 

414 

4,280 

5 

60 

2.98  x  0.5 

1.50 

400 

5,630 

6 

90 

j  5.86  x  0.99) 
\  3.00  x  i.  o  J 

3.00 

311 

9,600 

7 

90 

I  .  02  x  i  .  03 

1.05 

300 

9,750 

8 

1  2O 

3  .  oo  x  1.51 

4-53 

272 

10,170 

9 

60 

3.01  x  0.77 

2.31 

270 

12,970 

10 

60 

3.01  x  0.99 

2.99 

207 

18,070 

ii 

60 

5.84  x  i.  oo 

5-84 

207 

17,700 

12 

30 

2.99  x  0.50 

1.50 

206 

16,850 

13 

90 

3.00  x  1.53 

4-59 

204 

19,990 

14 

60 

1.03  X  1.02 

1.05 

200 

17,270 

15 

30 

3.01  x  0.76 

2.30 

135 

27,770 

16 

30 

3.00  x  i.  oo 

3-oo 

104 

29,660 

17 

30 

I  .  O2  X  I  .  O2 

i  .04 

100 

25,330 

18 

15 

I  .  O2  X  I  .  O2 

1.04 

50 

34,550 

J9 

7-5 

1.02  X  1.02 

1.04 

25 

48,680 

20 

3-3 

1.02  X  I  02 

1.04 

13 

50,400* 

Bore  this  without  failure. 


Tables  I.,  II.  and  III.  contain  the  results  of  some  English 
experiments  on  small  flat-end  wrought-iron  columns  of  different 


Art.  54.] 


FLAT  END   COLUMNS. 


475 


All  these  experiments  were  on  small   cross  sections.      In  reality  the  columns 
were  little  more  than  models. 


4/6 


GRAPHICAL  REPRESENTATION. 


.[Art.  54. 


forms  of  cross  section.  These  results  are  taken  from  the  "  Pro- 
ceedings of  the  Institution  of  Civil  Engineers,"  of  London,  Vol. 
XXX.  The  experiments  on  tubular  and  angle-iron  columns 
(Tables  I.  and  III.)  were  made  by  Mr.  Davies,  while  those  on 
solid  rectangular  columns  (Table  II.)  were  made  by  Mr.  Hodg- 
kinson.  The  graphical  representation  of  these  results  is  shown 
by  a  very  accurate  construction  in  Plate  III.  Fig.  I  belongs 
to  Table  I.  ;  Fig.  2  to  Table  II.  ;  and  Fig.  3  to  Table  III.  The 
result  shown  at  a  (No.  I  of  Table  II.),  Fig.  2,  is  most  anoma- 
lously high,  as  is  very  evident,  and  has  been  neglected. 

The  horizontal  scale  shows  the  ratio  of  /  over  r,  while  the 
vertical  scale  shows  pounds  per  square  inch,  to  a  scale  of 
30,000.00  pounds  to  the  inch. 


TABLE  III. 


3"  x  3"  x 


—Flat  Ends. 


ULT.  RESIST.   PER 

NO. 

LENGTH. 

AREA. 

/-*-  r. 

SQ.  IN. 

Inches. 

Sq.  Ins. 

Pounds. 

I 

Co     , 

I.7S 

71 

23,600 

2 

43 

1.78 

56 

29,480 

3 

36 

I.78 

42 

35,380 

4 

18 

1.78 

21 

39,400 

There  are  only  four  results  with  angle  irons,  but  so  far  as 
they  extend,  they  are  less,  for  the  same  value  of  /  over  r,  than 
those  for  either  tubes  or  solid  rectangular  sections.  This  was 
to  be  expected,  since  the  legs  of  angles  are  comparatively  thin 
and  give  very  little  lateral  support  to  each  other.  A  single 
unsupported  angle  iron,  therefore,  does  not  make  a  good  com- 
pression member. 


Art.  55.]  ANGLE  IRON  COLUMNS.  477 

These  results,  in  connection  with  those  of  Art.  51,  show 
very  clearly  that  an  empirical  curve  (or  formula)  may  be  con- 
structed to  cover,  with  sufficient  accuracy  for  practical  pur- 
poses, columns  of  different  forms  of  cross  section,  provided  they 
are  so  built  that  their  component  parts  are  mutually  supporting. 

As  compression  members  of  single  angle  irons  with  fixed 
ends  are  quite  common  in  some  riveted  bridge  and  roof  trusses, 
it  would  be  desirable  to  frame  a  formula  on  an  extended  series 
of  numerous  experiments.  In  the  present  instance  this  is  im- 
possible, but  the  following  formula  may  be  used  with  safety 
for  equal  legged  angle  iron  columns  with  flat  or  fixed  ends,  so 
long  as  /  -*•  r  lies  between  20  and  100: 

/  =  200,000  V/I          (I) 


in  which/  is  the  ultimate  resistance  per  square  inch.  An  ap- 
plication to  the  columns  of  Table  III.  gives  the  following  re- 
sults : 

No.                            I  •+-  r  p. 

1   71   23,740  Ibs.  per  square  inch. 

2   56 26, 760  Ibs.  per  square  inch. 

3     42  30,860  Ibs.  per  square  inch. 

4 21   43,600  Ibs.  per  square  inch. 

By  comparison  with  the  results  in  Table  III.,  the  deviations 
from  the  actual  resistances  given  by  experiment  may  be  seen 
at  a  glance. 

Art.  55. — Limit  of  Applicability  of  Euler's  Formula. 

The  great  range  of  /  -r-  r  in  the  experimental  results  of 
Tables  I.  and  II.  of  the  preceding  Article,  furnishes  means  of 
testing  the  applicability  of  Euler's  formula  with  high  values  of 
that  ratio. 


47  8  LIMIT  FOR  EULEKS  FORMULA.  [Art.  55. 

Mr.  Hodgkinson  determined  the  mean  value  of  the  com- 
pressive  coefficient  of  elasticity  for  some  wrought  iron  of  pre- 
sumably the  same  grade  as  that  to  which  Table  II.  belongs,  at 
about  23,250,000  pounds  per  square  inch.  That  value  gives  : 

47T2 E  —  917,920,000. 
Taking  /  -f-  r  from  No.  i,  Table  I. : 


/A2  i 

/  =  47t2E(-}    =  16,000  (nearly) 

Experiment  gave  14,670  J 

Taking  /  -f-  r  from  No.  7,  Table  II. : 

-)    =  10,200  (nearly)  1 


...    (2) 


•    •    •    (3) 
Experiment  gave  9,750 


Taking  /  -~  r  from  No.  5,  Table  II.  :. 

/  =  4^  =  5,740  (nearly) 


(4) 
Experiment  gave  5,630 

Taking  /  -r  r  from  No.  3,  Table  II.  : 

j)'  =3J50  (nearly) 
Experiment  gave  3,380 


Art.  56.]  REDUCTION  OF  END   SECTION.  479 

Taking  /  —  r  from  No.  2,  Table  II. : 

/  =  4*j£j^y  =  2,220  (nearly) 


....    (6) 
Experiment  gave  2,410 

In  Eq.  (2),  /  -r-  r  is  240,  yet  the  result  by  formula  is 
only  a  little  too  large.  With  I  -±  r  ranging  from  300  to  643, 
the  formula  gives  very  satisfactory  results.  These  tests  would 
seem  to  show,  therefore,  that  only  when  /  -f-  r  becomes  equal 
to  about  250  for  flat-end  columns,  does  Ruler's  formula  become 
applicable  to  wrought-iron  compression  members,  but  that 
above  that  limit  it  gives  very  satisfactory  results. 

This  is  an  interesting  and  striking  confirmation  of  the  cor- 
rectness of  the  formula,  which,  as  was  stated  in  Art.  25,  is 
based  on  the  supposition  that  the  lateral  dimensions  are  very 
small  compared  with  the  length. 

Art.  56. — Reduction  of  Columns  at  Ends. 

When  columns  are  built  of  angle  irons,  channel  bars,  or  I 
beams,  it  is  frequently  the  practice  to  cut  off,  for  some  distance 
back  from  the  ends,  the  flanges  of  bars  or  beams,  or  one  of  the 
legs  of  angle  irons,  in  order  to  give  clearance  for  other  mem- 
bers of  the  structure.  In  such  cases  the  whole  compression  to 
which  the  column  is  subjected  is  carried,  at  the  ends,  by  the 
webs  of  the  bars  or  beams,  or  legs  of  the  angles,  which  are  thus 
solid  rectangular  columns  of  great  comparative  breadth  and 
.little  thickness,  even  when  reinforced  by  plates  of  the  same 
thickness  as  the  webs  or  legs.  In  such  cases,  the  angle  iron 
experiments  of  Mr.  Davies  (a  part  of  which  are  given  in  Art. 
54),  and  a  most  valuable  set  of  full  sized,  latticed,  channel- 
bar  column  tests,  made  at  the  works  of  the  Keystone  Bridge 
Co.,  Pittsburgh,  Penn.  ("  The  American  Engineer,"  4th  Feb., 


4^0  TIMBER   COLUMNS.  [Art.  57. 

1882),  show  that  the  full  resistance  of  the  column  is  not  devel-' 
oped,  but  that  they  fail  at  the  ends  where  the  cutting  away  of 
the  flanges  and  legs  reduces  the  column  to  two  thin,  weak,  rect- 
angular columns.  Columns,  therefore,  should  never  'be  cut 
away  in  the  manner  indicated  unless  the  circumstances  render 
it  absolutely  necessary,  and  then  the  ends  should  be  reinforced 
by  extraordinarily  heavy  thickening  plates,  so  that  the  sum  of 
the  resistances  of  these  rectangular  columns,  at  each  end,  shall 
be  equal  to  that  of  the  column  as  a  whole. 

Art.  57.— Timber  Columns. 

Tests  of  this  class  of  members,  the  results  of  which  have 
been  published,  although  of  great  value,  have  not  been  made 
with  sufficiently  large  ratios  of  length  to  radius  of  gyration  to 
produce  true  "  long  column  "  failures.  This  renders  impos- 
sible the  establishment  of  a  long  column  formula  or  diagram 
for  practical  use  in  connection  with  the  use  of  long  timber 
columns. 

Some  very  valuable  experiments,  however,  have  been  made 
with  full  sized  columns  having  lengths  as  great  as  fourteen 
feet.  The  first  results  to  be  given  are  those  of  a  large  number 
of  tests  by  Prof.  Lanza,  of  Boston,  in  which  he  used  the  United 
States  testing  machine  at  Watertown,  Mass.  These  tests  were 
made  during  1881,  on  such  members  as  are  commonly  used  in 
the  construction  of  cotton  and  woollen  mills. 

Table  I.  contains  the  results  of  Prof.  Lanza's  tests.  A  large 
majority  of  the  columns  had  cores  bored  out  of  the  centre, 
which  varied  in  diameter  from  1.5  to  2.0  inches.  The  ab- 
sence of  material  did  not  affect,  in  any  way,  so  far  as  could  be. 
observed,  the  resistance  per  square  inch. 

Column  20  had  the  force  applied  2*/£  inches  out  of  centre  at 
one  end,  and  column  35,  1.9  inches.  These  tests  were  made  in 
order  to  observe  the  effect  of  eccentricity  in  the  application  of 
loads.  They  show  a  marked  decrease  in  ultimate  resistance. 


Art.  57.] 


LANZA'S  EXPERIMENTS. 


481 


TABLE    I. 
Timber  Mill  Columns. 


NO. 

FORM. 

LENGTH, 
FEET. 

DIA.  IN  INCHES  AT 

SECT.   AREA    IN 
SQ.   INS.  AT 

ULT.  RESIST., 
LBS.  PER  SQ. 
INCH. 

ENDS. 

Top. 

Mid. 

Base. 

Top. 

Base. 

Yellow  pine  ;  partially  seasoned. 

I 

Round  

12.  OI 

9-31 

IO.65 

10.55      65.89 

85-22 

4,098 

Flat 

2 

12.  02 

Q    TT 

jO.  O7 

ci  84. 

77-36 

3 

11 

12.01 

7-54 

08 

8-99 

5  j  -04 

42-38 

61.21 

4<Vi9 

4 
5 

H 

12.02 
12  00 

io'4° 

7.80 

7-79 

10.45 

29.98 
83-75 

45-47 

4,602 
4,657 

Cylindrical.. 

6 

u 

II  92 

8.96 



8.96 

61.19 



4,086 

7 

" 

7.70 



7.70 

44.72 



4,584 

8 

H 

2.OI 

10.46 



10.46 

85.92 



4,422 

9 

2.00 

9.98 



9.98 

78.23 



4,705 

10 

2.01 

8.91 



8.91 

62.35 



4-33° 

ii 

U 

2.OO 

7-79 



7-79 

47.66 



4.5" 

12 

Square  

12.44 

8-43 



8.40 

68.80 



1 

13 

Cylindrical.. 

"•57 

11.93 

8.30 
9.92 

— 

8-30 
9.92 

63.10 

75-45 



3$o4 
3,5*2 

M 

Yellow  pine  ;  air  seasoned. 

3 

Round  .  . 

13-99 

2.OO 

7.70 
7.70 



7.90 
7.90 

44-56 

47.01  1     4,488 
1     4,892 

tl 

Cylindrical.. 

Yelloiv  pine  ;   dock  seasoned. 

19 

Cylindrical  .  . 
Square  

11.92 
1.98 

TI.QO 

8.00 

7-93 
8.08 



8.00 
7.98 
8.08 

48.26 
48.00 
63.28 



4,662 
3,604 

•3    ,177 

One  flat  :  one  round. 
Flat. 
One  flat  ;  one  round. 

20 

*    *Vo 

12.94 

8-75 



8.92 

76.04 



O,4// 
3,682 

One  flat  ;  one  round. 

21 

u      

12.85 

10.05 



10.13 

99-79 



5-  "I 

Flat,  pintle. 

22 
23 

lt        

2.OO 
2.OI 

3-98 

IO.2O 



9.02 
10.07 

81.00 
icv>.i  r 



5,951 

Flat. 

White  ivood 

•  partially  seas 

oned. 

24 

Round  

12.  OI 

8.46 

9.61 

9.65 

54.02  1  70.95 

3,333 

« 

25 



11-97 

6.38 

7.40 

7.72 

29.78  |  44.62 

2,687 

White  oak  :  partially  seasoned. 

26 

3 

Round  

12.01 
12.01 
12.01 

9-13 

8-37 
7-55 

10.15 

9.40 
8-75 

1  1.  01 

10.23 
9-05 

63.28 

52-83 
42.58 

39.01 
80.00 
62.14 

3,003 
3,786 
3,758 

: 

29 
3° 

Cylindrical.. 

12.01 
12.  OO 

6.60 

IO.OO 

7-79 

8.06 

32.02 
7670 

48.83 

3,435 
2,478 

• 

31 

6-33 

10.00 

—— 

— 

76.70 



" 

32 

.. 

2.OO 

9-98 





78.23 



3,  '32 

M 

33 

2  00 

8.18 



• 

52-55 



3,M° 

1 

34 

1.98 

7-73 





4^-93 



3,303 

u 

35 

"•93 

8.20 

50.92 

1,964 

482 


TIMBER   COLUMNS. 


[Art.  57. 


TABLE   I.— Continued. 


SECT.  AREA  IN 

jTJ 

DIA.  IN  INCHES  AT 

*-*       ££ 

LENGTH, 

SQ.  INS.  AT 

w  S 

NO. 

FORM. 

"  g  5 

ENDS. 

FEET. 

5  3  2 

Top. 

Mid. 

Base. 

Top. 

Base. 

White  oak  ;  used  in  mill  6%  years. 

36 

Round     .   .. 

12.08 

5-85 



6.84 

23-89 

33-76 

4,604 

Flat. 

44 

12.08 

5.85 



6.85 

24.05 

34-02 

6,029 

4i 

38 

44 

I2.II 



6.70 

23.92 

32.12 

4,680 

41 

39 

44 

12.05 

6.  02 



6-75 

25.47 

32-79 

2,945 

ii 

40 

44 

11.72 

6.10 



6.83 

26.08 

33-50 

ii 

44 

12.01 

5-97 



6.74 

25.09 

32.78 

4,225 

ii 

42 

12.07 

5-75 

6.88 

22.98 

34-19 

3*264 

White  oak  ;   used  in  will  25  years. 

43 

Cylindrical 

13-87 

10.56 





84-74 



4,602 

" 

44 

14.00 

10.54 





84.83 



4-951 

II 

45 

44 

13.89 

10.54 





84.40 



4,266 

" 

46 
47 

ii 

13.80 
13.92 

10.50 

10.20 





83-75 
79-T7 



3,881 

II 

Flat,  pintle. 

48 

u 

13.89 

10.  80 





82.68 



4,'838 

•*                  Ii 

49 
So 

Round  
Cylindrical 

13-67 
1385 

9-25 

9-55 

— 

9-50 

64.36 
68.65 

68.04 

3,434 
4,618 

Flat. 

M 

Si 

52 
53 

Round  

13-65 
13.90 
11.51 

9.40 
9-35 
5-98 

— 

7.20 

66.56 
65.82 
26.03 

38.66 

3,98! 
3.266 
6,147 

Pintle  ends. 
Flat. 

White  oak  ;  thoroughly  seasoned:  i  year  old. 

54 

3 

Cylindrical  .  . 

12.00 
II.  12 

2.00 

7-74 
10.95 
10.91 

I 



45-04 
91.16 

= 

1^865 
4,450 

Flat. 
One  flat  ;  one  round. 
Flat. 

Although  the  ends  of  Nos.  39  and  44-51  were  flat,  they 
were  not  parallel. 

All  the  columns  indicated  by  "Round"  were  tapered,  and 
they  almost  invariably  gave  way  by  the  crushing  of  the  fibres 
at  the  small  end.  In  all  such  columns  the  ultimate  resistance 
is  per  square  inch  of  the  small  end. 

Some  of  the  square  columns  had  their  corners  slightly 
beveled. 

In  his  report  to  the  Boston  Manufacturers'  Mutual  Fire  Ins. 
Co.,  Prof.  Lanza  says:  "The  immediate  location  of  the  fract- 
ure was  generally  determined  by  knots;"  .  .  .  but  states 


Art.  57.] 


LAIDLEY'S  EXPERIMENTS. 


483 


that,  whether  knotty  or  straight  grained,  failure  took  place  in 
the  tapered  columns  at  the  small  ends.  Tapering  a  column, 
therefore,  to  the  extent  shown  in  these  cases,  is  a  source  of 
weakness. 

TABLE  II. 
Yellow  Pine. 


NO. 

LENGTH, 
INCHES. 

FORM   OF  SECTION. 

SECTION    DIMENSIONS, 
INCHES. 

ULTIMATE 
ANCE   PER 

RESIST- 
SQ.   IN. 

Lbs. 

I 

20.4 

*  Circular. 

10.2  Diam. 

6,676  ' 

8 

2 

119-95 

Square. 

II           X     II 

6,230 

"8 

3 

119.90 

II          X     II 

6,552 

4> 

4 

20.0 

10.4  x   10.4 

7,936 

o 

5 

16.0 

• 

8x8 

8,165 

i 

6 

8.0 

4x4 

7,394 

V) 

7 

3-0 

1.5  x     1-5 

5,533 

*o 
C   en 

8 

6.0 

3x3 

8,644 

^rts 

9 

6.0 

3x3 

8,133 

•S& 

10 

3-o 

1-5  x     1.5 

8,389 

a 

ii 

3-o 

1.5  x     1.5 

8,302 

1 

12 

3-o 

i-5  x     1.5 

6,355 

13 

14.0 

4.6  x     4.6 

9,947 

^ 

14 

17.2 

4.6  x     4.6 

10,250 

% 

15 

19.1 

5-3  x     5.3 

7,820  , 

& 

16 

180.0 

Rectangular. 

16       x   13.65 

3,070 

17 

180.0 

16.2  x     7.0 

2,795 

18 

180.0 

17       x     8.75 

3,180 

Nos.  13,  14  and  15  were  pine  of  very  slow  growth. 
Nos.  16,  17  and  18  were  very  green  and  wet.. 

Tables  II.  and  III.  contain  the  results  of  Col.  Laidley's 
tests,  some  of  which  belong  to  short  blocks.  These  tests  were 
made  during  1881,  and  a  detailed  account  of  them  is  given  in 
"Ex.  Doc.  No.  12,  4/th  Congress,  1st  Session." 

These  experiments  give  some  very  important  deductions. 

In  the  first  place,  within  the  limits  of  the  ratio  of  length  to 
diameter,  or  shortest  side  of  rectangular  section,  appearing  in 
these  tests,  the  ultimate  resistance  is  essentially  independent 


TIMBER    COLUMNS. 


[Art.  57. 


TABLE    III. 

Spruce,  thoroughly  seasoned. 


NO. 

LENGTH, 
INCHES. 

FORM    OF   SECTION. 

SECTION    DIMENSIONS, 
INCHES. 

ULTIMATE    RESIST- 
ANCE PER   SQ.  IN. 

Lbs. 

I 

24 

Rectangular. 

5-4 

x  5-4 

4,946 

2 

24 

5-4 

x  5-4 

4,811 

3 

36 

5-4 

x  5-4 

4,874 

4 

36 

5-4 

x  5-4 

4,500 

5 

60 

5-4 

x  6.4 

4,451 

6 

60 

5-4. 

x  6.4 

4,943 

7 

1  2O 

5-4 

x   5-4 

3,967 

8 

I2O 

5-4 

x   5-4 

4,908 

9 

60 

5-4 

x  5-4 

5,275 

10 

30 

5-4 

x   5-4 

5,372 

ii 

15 

5-4* 

x   5-4 

5,754 

12 

121.  2 

Circular. 

12.4 

Diam. 

4,681 

of  the  length.  This  is  the  result  of  the  action  of  causes  noticed 
in  the  consideration  of  wrought-iron  columns  composed  of  C's. 
The  ultimate  resistance  of  any  such  column,  therefore,  is  to  be 
obtained  by  multiplying  the  area  of  its  cross  section  by  the 
ultimate  resistance,  per  square  inch,  of  short  blocks. 

In  Prof.  Lanza's  experiments,  the  greatest  ratio  of  length 
to  radius  of  gyration  was  about  86.  Below  this  value  the  gen- 
eral conclusion  just  given  may  be  expected  to  hold,  but  prob- 
ably not  much  above  it. 

In  Col.  Laidley's  tests  the  greatest  value  of  the  same  ratio 
was  about  90  (No.  17  of  Table  II.),  at  which  there  seemed  to 
be  a  little  decrease  in  ultimate  resistance. 

Again,  it  is  to  be  observed  that  Prof.  Lanza's  results  are 
much  less  than  those  of  Col.  Laidley  for  the  same  timber.  The 
columns  of  the  former  were  of  ordinary  merchant  material, 
with  the  usual  accompaniment  of  knots,  weak  spots,  crooked 
grain,  etc.,  while  the  latter  experimented  with  fine,  straight- 
grained  timber. 


Art.  57].  C.    SHALER   SMITH'S  FORMULA.  485 

The  slow  growth  specimens  (13,  14  and  15,  of  Table  II.), 
gave  much  the  highest  results,  while  the  wet  and  unseasoned 
ones  (16,  17  and  18)  gave  the  lowest  of  all. 

Hence,  the  ultimate  resistance  of  timber  columns  will  de- 
pend upon  quality  and  condition  of  material,  mode  of  growth, 
degree  of  seasoning,  etc.,  etc. 

Table  II.  also  shows,  what  has  been  observed  elsewhere, 
that  smaller  specimens  give  higher  results  than  larger  ones. 


Formula  of  C,  Shaler  SmM,  C.  E. 

During  the  winter  of  1861-62,  Mr.  C.  Shaler  Smith  con- 
ducted a  series  of  over  1,200  tests  of  full  size  yellow  pine 
square  and  rectangular  columns  for  the  Ordnance  Dept.  of  the 
Confederate  Government.  The  results  of  these  tests  have 
never  been  published,  but  Mr.  Smith  has  kindly  furnished  the 
writer  with  the  following  summary : 

The  tests  were  grouped  as  follows : 

"  1st.  Green,  half-seasoned  sticks  answering  to  the  specifi- 
cation, '  good  merchantable  lumber.' 

"  2d.  Selected  sticks  reasonably  straight,  and  air-seasoned 
under  cover  for  two  years  and  over. 

11  3d.  Average  sticks  cut  from  lumber  which  had  been  in 
open  air  service  for  four  years  and  over." 

If  /  =  length  of  column  in  inches  ; 

d  =  least  side  of  column  section  in  inches  ; 
and       /  =  Ult.  Comp.  resistance  in  Ibs.  per  sq.  in.  ; 

then  the  formulae  found  for  these  three  groups  were : 

For  No.  i  :      /  =  - 

1  +  2^^ 


486  TIMBER    COLUMNS.  [Art.  57. 


T-,        AT  8,200 

For  No.  2  :      /  =  - 

1  / 


^  300  d2 

For  No.  3  :      /  =  -     5>O^°  ^  . 

250  ^ 

But  in  order  to  provide  against  ordinary  deterioration  while 
in  use,  as  well  as  the  devices  of  unscrupulous  builders,  Mr. 
Smith  recommends  the  formula  for  group  No.  3  as  the  proper 
one  for  general  application.  He  also  recommends  that  the 

factor  of  safety  shall  be  ,*  /-  until  25  diameters  are  reached, 


and  five  thenceforward  up  to  60  diameters.     This  last  limit  he 
regards  as  the  extreme  for  good  practice. 

Mr.  Trautwine  computed  his  tables  from  tests   of  group 
No.  3. 


CHAPTER    VIII. 

SHEARING  AND  TORSION. 

Art.  58.— Coefficient  of  Elasticity. 

It  has  already  been  shown  in  some  of  the  Articles  of  the 
first  portion  of  this  book,  on  shearing  and  torsion,  that  the  co- 
efficients of  elasticity  for  those  two  stresses  are  the  same  ;  and, 
indeed,  that  those  two  stresses  are  identical  in  character.  The 
coefficients  of  elasticity,  given  in  this  Article,  are  then  derived 
chiefly  from  experiments  in  torsion. 

In  his  "  Legons  de  M£canique  Pratique,"  1853,  Gen.  Arthur 
Morin  gives  the  following  rtsumt  of  the  results  of  experiments 
up  to  that  time,  in  which  G  is  the  coefficient  of  elasticity,  for 
shearing,  in  pounds  per  square  inch. 

MATERIAL.  G.,  Ibs. 

Soft  wrought  iron 8,571,000 

Iron  bars 9,523,000 

German  steel 8,571,000 

Fine  cast  steel 14,300,000 

Cast  iron 2.857,000 

Copper 6,237,000 

Bronze 1,523,000 

Oak 571,400 

Pine 618,600 

The  above  value  for  cast  iron  must,  however,  be  much  too 
small,  as  will  presently  be  seen. 

In  "  Der  Civilingenieur,"  Heft  2,  1881,  the  results  of  some 
very  interesting  and  important  experiments  on  cast-iron  rods 
or  prisms  of  various  cross  sections,  by  Prof.  Bauschinger,  are 


488 


SHEARING  AND    TORSION. 


[Art.  58. 


given  in  full  detail.  The  rods  or  prisms  were  about  40  inches 
long,  and  were  subjected  to  torsion,  while  the  twisting  of  two 
sections  about  20  inches  apart,  in  reference  to  each  other,  was 
carefully  observed.  The  results  for  four  different  cross  sections 
will  be  given — /.  ^.,  circular,  square,  elliptical  (the  greater  axis 
was  twice  the  less)  and  rectangular  (the  greater  side  was  twice 
the  less).  In  each  case  the  area  of  cross  section  was  about 
7.75  square  inches.  The  angle  a  is  the  angle  of  torsion — i.  e., 
the  angle  twisted  or  turned  through  by  a  longitudinal  fibre, 
whose  length  is  unity,  and  which  is  at  unit's  distance  from  the 
axis  of  the  bar. 

G. 

0.007  degree 7,466,000  Ibs.  per  sq.    n. 


SECTION. 

Circular  

a. 

jo.  007 
(0.07 

Elliptical 

30.009 

(  0.076 

Square                 . 

j  0.008 

\  0.073 

Rectangular  

j     O.OI 

\  0.08 

6,157,000 
7,437,000 
6,228,000 
7,039,000 
5,987,000 
6,996,000 
5,716,000 


The  formula  by  which  G  is  computed,  when  the  torsional 
moment  and  angle  a  are  given,  is  the  following : 


a 


0) 


in  which  M  is  the  twisting  moment ;  A  the  area  of  the  cross 
section ;  Ip  the  polar  moment  of  inertia  of  that  cross  section  ; 
and  c  a  coefficient  which  has  the  following  values  : 

4?r2  =  39.48  for  circle  and  ellipse ; 
42.70   "    square ; 
42.00   "    rectangle ; 

as  was  shown  in  Art.  10. 

Bauschinger's    experiments   show   that   the    coefficient    of 


Art.  58.] 


COEFFICIENTS  OF  ELASTICITY. 


489 


shearing  elasticity  for  cast  iron  may  be  taken  from  6,000,000  to 
7,000,000  pounds  per  square  inch  ;  also,  that  it  varies  for  differ- 
ent ratios  between  stress  and  strain. 

It  has  been  shown  in  Art.  4,  that  if  E  is  the  coefficient  of 
elasticity  for  direct  stress,  and  r  the  ratio  between  direct  and 
lateral  strains,  for  tension  and  compression,  that  G  may  have 
the  following  value : 


G  = 


(2) 


Prof.  Bauschinger,  in  the  experiments  just  mentioned, 
measured  the  direct  strain  for  a  length  of  about  4.00  inches, 
and  the  accompanying  lateral  strain  along  the  greater  axis  of 
the  elliptical  and  rectangular  cross  sections,  and  thus  deter- 
mined the  ratio  r  between  the  direct  and  lateral  strains  per 
unit,  in  each  direction.  The  following  were  the  results : 


Compression . 


G. 


Circular 

Elliptical 

Square 

Rectangular. . . 


. ..  0.22  6,541,000  Ibs.  per  sq.  in. 

...  0.23 6,541,000    "     "     "     " 

. ..  0.24 6,442,000    "     "     "     " 

. ..  0.24 6,499,000    "     "     "    " 


Tension. 


Circular 

Elliptical , 

Square 

Rectangular. . . 


0.23 
0.21 
0.26 

0.22 


6,570,000  Ibs.  per  sq.  in. 
6,811,000  "  "  "  " 
6,399,000  "  "  "  " 
6,527,000  "  "  "'  " 


The  values  of  E  are  not  reproduced,  but  they  can  be  calcu- 
lated indirectly  from  Eq.  (2)  if  desired. 

It  is  seen  that  the  values  of  G,  as  determined  by  the  differ- 
ent methods,  agree  in  a  veiy  satisfactory  manner,  and  thus  fur- 
nish experimental  confirmation  of  the  fundamental  equations 
of  the  mathematical  theory  of  elasticity  in  solid  bodies. 


49°  SHEARING  AND    TORSION.  [Art.  59. 

The  fact  that  G  is  essentially  the  same  for  all  sections  is  also 
strongly  confirmatory  of  the  theory  of  torsion,  in  particular. 

These  experiments  show  that,  for  cast  iron,  the  lateral  strains 
are  a  little  less  than  one  quarter  of  the  direct  strains.  If  rwere 
one  quarter,  then  G  —  f  E ;  or  E  =  f  G. 

In  the  "  Journal  of  the  Franklin  Institute,"  for  1873,  Prof. 
Thurston  gives  the  following  values  of  G,  as  determined  from 
experiments  with  his  torsion  machine. 

White  Pine G  =  220,000  pounds  per  sq.  in. 

Yellow  Pine,  sap G  =  495,000  "  "  "  " 

Yellow  Pine,  heart G=  495,000  "  "  "  " 

Spruce G  =  211,000  "  "  "  " 

Ash G  =  410,000  "  "  "  " 

Black  Walnut G—  582,000  "  "  "  " 

Red  Cedar G  —  890,000  "  "  "  " 

Spanish  Mahogany G  =  660,000  "  "  "  " 

Oak G  =  570,000  "  "  "  " 

Hickory G  =  910,000  "  "  "  " 

Locust £  =  1,225,000  "  "  "  '* 

Chestnut G  =  355,000  "  '*  "  '* 

The  specimens  were  small  ones,  and  the  timber  was  sea- 
soned. 

Art.  59.— Ultimate  Resistance. 

Before  considering  the  ultimate  shearing  resistance  of  spe- 
cial materials  it  will  be  well  to  notice  the  two  different  methods 
in  which  a  piece  may  be  ruptured  by  shearing. 

If  the  dimensions  of  the  piece  in  which  the  shearing  force 
or  stress  acts  are  very  small,  i.e.,  if  the  piece  is  very  thin,  the 
case  is  said  to  be  that  of  "  simultaneous  "  shearing.  If  the 
piece  is  thick,  so  that  those  portions  near  the  jaws  of  the  shear 
begin  to  be  separated  before  those  at  some  distance  from  it, 
the  case  is  said  to  be  that  of  "  shearing  in  detail."  In  the  lat- 
ter case  failure  extends  gradually,  and  in  the  former  takes  place 
simultaneously  over  the  surface  of  separation.  Other  things 


Art.  59.]  ULTIMATE  RESISTANCE.  49 1 

being  the  same,  the  latter  case  (shearing  in  detail),  will  give 
the  least  ultimate  shearing  resistance  per  unit  of  the  whole 
surface. 

In  reality  no  plate  used  by  the  engineer  is  so  thin  that  the 
shearing  is  absolutely  simultaneous,  though  in  many  cases  it 
may  be  essentially  so. 


Wrought  Iron. 

The  following  averages  (each  result  being  an  average  of 
six  tests),  are  from  Chief  Engineer  Shock's  experiments,,  in 
1868,  on  ordinary  commercial  "  rounds  "  ("  Steam  Boilers,"  by 
William  H.  Shock,  Chief  Engineer,  U.  S.  N.),  in  which  5  is  the 
ultimate  shearing  resistance  in  pounds  per  square  inch  : 

s. 


DIAM.    OF    ROUND.  SINGLE   SHEAR.  DOUBLE   SHEAR. 

o.  5      inch 44,150  Ibs 41,090  Ibs. 

0.625  inch 39,250  Ibs 38,670  Ibs. 

0.75    inch 39, 550  Ibs 39,770  Ibs. 

0.875  inch 41, 500  Ibs 37,890  Ibs. 

i. oo    inch 40, 700  Ibs 37,650  Ibs. 


Although  these  figures  show  some  irregularities,  the  general 
result  is  unmistakable,  and  shows  a  decrease  of  5  with  an  in- 
crease of  diameter. 

The  results  of  experiments  at  Bristol,  England,  by  Mr. 
Jones  ("Proc.  Inst.  Mech.  Engrs.,"  1858),  on  punching  plate 
iron,  are  as  follows : 

THICKNESS   OF   PLATE.  DIAM.   OF  HOLE.  S. 

0.437  inch 0.250 54,700  Ibs.  per  sq.  in. 

0.625     "    0.500 60,900    "      "     "     " 

0.625     "    0.750 52,900    "      "     "     " 

0.875     "    0.875 51,700    "      "    "     " 

i. ooo    "    i. ooo 55,100   "      "    "     " 


492  SHEARING  AND    TORSION.  [Art.  59. 

Mr.  C.  Little  found  the  following  for  English  "  hammered 
scrap  bars  and  rolled  iron,"  with  parallel  cutters  or  shears  : 

AREA  CUT.  DIRECTION.  S. 

0.50  x  3.00  ins Flat 49,950  Ibs.  per  sq.  in. 

0.50  x  3. oo  ins Edge 5I.75O  "  "  "  " 

.00  x  3.00  ins Flat 5!,75O  "  "  "  " 

.00  x  3. coins Edge 50,850  "  "  "  " 

.00  x  3.02  ins Flat 44,350  "  "  "  " 

.00  x  3.02  ins Edge 46,150  "  "  "  " 

.80  x  5.00  ins Edge 46,150  "  "  "  " 

In  these  experiments  the  edges  of  the  shears  were  always 
parallel  to  each  other,  thus  tending  to  produce  simultaneous 
shearing.  In  ordinary  workshop  practice,  however,  the  jaws  of 
the  shears  make  a  constant  angle  with  each  other,  thus  shear- 
ing successive  portions  of  the  material  as  the  jaws  approach, 
whatever  may  be  the  dimensions  of  the  piece,  and  conse- 
quently always  producing  shearing  in  detail.  In  the  experi- 
ments (by  the  same  authority,  i.  e.,  Mr.  C.  Little,  "  Proc.  Inst. 
Mech.  Engrs.,"  1858)  from  which  the  following  results  were 
deduced,  the  angle  between  the  jaws  of  the  shears  was  an  incli- 
nation of  i  in  8  : 


BARS.  FLATWAYS.  EDGEWAYS. 


3  X 

4-5  x 
3.0  x 
5-25  x 
6.00  x 


,5      ins 5=40,800 45,000  Ibs.  per  sq.  in. 

.375  ins 5=32,000 40,100     "     "    "     " 

.00    ins 5=35,200 47,300     "     "    "     " 

•  75    ins 5=37,400 50,600    "     "    "     " 

.50    ins 5=33,600 41,200     "     "    "     " 


As  was  to  be  expected,  the  "Edgeways"  results  are  much 
the  largest,  as  with  that  position  of  the  bar  the  shearing  ap- 
proached more  nearly  the  simultaneous  condition.  These 
results  show  that  it  is  much  more  economical  to  shear  a  bar 
flatways  than  edgeways. 

Mr.  Edwin  Clark  ("  On  the  Tubular  Bridges  ")  found  the 
resistance  of  ^-inch  rivet  iron,  in  single  and  double  shear,  to 


Art.  59.]  ULTIMATE  RESISTANCE.  *  493 

vary  from  49,500  to  54,100  pounds  per  square  inch.  The  ten- 
sile resistance  of  the  same  iron  was  about  53,800  pounds  per 
square  inch. 

Reviewing  all  these  results,  the  ultimate  shearing  resistance 
of  wrought  iron  may  safely  be  taken  at  0.8  of  its  tensile  resist- 
ance, as  stated  by  Mr.  D.  K.  Clark. 


Cast  Iron. 

Very  few  experiments  on  the  resistance  of  cast  iron  to 
shearing  have  been  made,  as  this  metal  is  seldom  or  never  used 
to  resist  such  a  stress. 

Mr.  Bindon  B.  Stoney  ("Theory  of  Strains  in  Girders  and 
Similar  Structures,"  p.  357  of  2d  Edit.)  has  found,  by  experi- 
ment, that  the  ultimate  shearing  resistance  of  the  cast  iron 
with  which  he  experimented  varied  from  about  17,900  to 
20,200  pounds  per  square  inch.  He  concluded  that  the  shear- 
ing and  tensile  resistances  might  be  taken  the  same. 


Steel. 

In  1866  Mr.  Kirkaldy  tested  the  ultimate  shearing  resist- 
ance of  hematite  bar  steel  made  by  the  Barrow  Hematite  Steel 
Company,  of  Great  Britain,  and  found  the  mean  of  four  experi- 
ments with  hammered  cast  steel  rounds  1.25  inches  in  di- 
ameter, to  be  about 

56,500.00  pounds  per  square  inch. 

The  tensile  resistance  of  the  same  steel  was  found  to  be 
about  four-thirds  as  much. 

The  same  experimenter  investigated  the  ultimate  shearing 
resistance  of  four  grades  of  Fagersta  steel,  and  the  following 
results  are  taken  from  "  Experimental  Enquiry  into  the  Me- 


494  SHEARING.  [Art.  59. 

chanical  Properties  of  Fagersta  Steel/'  by  David  Kirkaldy, 
1873.  The  test-piece,  in  each  case,  was  turned  from  a  2-inch 
square  bar,  to  a  diameter  of  1.128  inches,  and  each  result  is  a 
mean  of  three  experiments.  5  is  the  ultimate  resistance  to 
shearing,  in  pounds  per  square  inch ;  r  is  the  ratio  of  ultimate 
shearing  over  ultimate  tensile  resistance  of  the  same  steel ; 
while  " d"  is  the  detrusion  or  relative  movement  of  one  part  of 
the  specimen  in  respect  to  the  other  at  the  instant  of  separa- 
tion over  the  entire  surface. 

MARK.  6".  r.  d. 

1.2 61,400.00  Ibs 0.73 o.iginch. 

0.9 79,740.00  "     0.75 0.25     " 

0.6 71,650.00  "    0.70 0.28    " 

0.3 45,410.00  "     0.74 0.32    " 

As  is  evident,  the  lower  "Mark"  numbers  belong  to  the 
softer  steels. 

In  each  case  two  surfaces  were  sheared,  as  the  "round" 
was  a  pin  for  three  links,  two  of  which  pulled  one  way,  and 
one  the  other. 

All  of  Mr.  Kirkaldy's  experiments  seem  to  show  that  the 
ultimate  shearing  resistance  of  steel  is  about  three-quarters  the 
tensile. 

Table  I.  contains  the  results  of  the  experiments  of  Prof.  A. 
B.  W.  Kennedy,  as  given  in  "Engineering"  for  May  6,  1881. 
The  tensile  resistance  of  the  same  steel  was  given  in  the  chap- 
ter on  "  Tension." 

The  specimens  were  round  and  of  mild  rivet  steel.  The 
ratio  of  the  ultimate  resistance  to  shearing  over  that  to  tension 
varied  from  0.80  to  0.89. 

In  the  "  Journal  of  the  Franklin  Institute,"  for  March,  1881, 
Charles  B.  Dudley,  Ph.D.,  gives  the  results  of  192  tests  of  rail 
steel,  the  specimens,  0.625  inch  round,  having  been  taken  from 
rails  which  had  been  subjected  to  service  for  considerable  pe- 
riods of  time  on  the  Penn.  R.  R.  The  tests  were  made  by 


Art.  59.] 


STEEL  AND   COPPER. 


495 


TABLE   I. 

Rivet  Steel. 


DIAMETER   IN   INCHES. 

ULTIMATE   RESIST.    IN 
LBS.    PER   SQ.    IN. 

MEAN. 

RATIO   OF    ULT.  SHEAR 
OVER    ULT.    TENSION. 

I.OO 

54,no 

I.OO 

54,930 

I.OO 
I.OO 

55,240 

52,830 

54,550 

0.89 

I.OO 

56,660 

I.OO 

53,530 

0.62 

60,260 

0.62 

59,400 

0.62 

59,6°o 

59,640 

O.S? 

0.62 

59,220 

0.62 

59,740 

o  62 

53,670 

0.62 

51,290 

0.62 

52,670  f- 

52,450 

0.80 

0.62 

53,ooo 

0.62 

5i,62oJ 

Mr.  J.  W.  Cloud,  engineer  of  tests   for  the   Penn.  R.  R.  Co. 
The  following  is  a  summary  of  the  results : 

{63,560.00  pounds  per  sq.  in.  (greatest). 
59,880.00  pounds  per  sq.  in.  (mean). 
53>38o.oo  pounds  per  sq.  in.  (least). 

The  percentages  of  carbon  and  ultimate  tensile  resistances 
are  given  in  Table  IV.  of  Art.  34.  By  reference  to  that  table 
it  will  be  observed  that  5  is  not  far  from  three-fourths  the 
tensile  resistance. 

Copper. 

From  some  English  experiments,  Mr.  Bindon  B.  Stoney 
concluded  that  the  ultimate  shearing  resistance  of  copper  was 
about  two-thirds  of  that  of  wrought  iron. 


496 


SHEARING. 


[Art.  59. 


Timber. 

In  treating  the  shearing  resistance  of  timber,  it  is  very  nec- 
essary to  consider  whether  the  shearing  takes  place  along  the 
fibres,  or  in  a  direction  normal  to  them. 

TABLE   I. 

Along  Fibres. 


KIND    OF   WOOD. 

GREATEST. 

MEAN. 

LEAST. 

Georgia  Pine  

*                         Q34 

843 

713 

White  Pine     . 

eoo 

482 

433 

Locust  

I^So 

I    16s; 

070 

White  Oak 

I  474 

I    25O 

I  O76 

Spruce  

647 

e  42 

46^ 

4IO 

760 

322 

TABLE   II. 
Across  Fibres. 


KIND    OF   WOOD. 

S. 

KIND   OF   WOOD. 

61. 

Ash 

6  280 

Locust      

7,176 

Beech            

5  223 

Maple  

6,355 

Birch  

e  CQC 

Oak,  white  

4,425 

I  372  to  I  5IO 

Oak   live        

8  480 

Cedar  Central  Amer  . 

3  4IO 

Pine   white      

2,480 

Cherry 

2  Q45 

Pine    yellow  northern.  . 

4,34O 

Chestnut             

I  535 

Pine    yellow,  southern  .  . 

5,735 

Dogwood  .        

6.5IO 

Pine,  yel  ,  very  resinous. 

5,053 

Ebony* 

7  7eo 

Poplar  

4,418 

Gum                        •  •    .  . 

e  8oO 

3,255 

Hemlock       

2.  75O 

Walnut,  black  

4,728 

6  045  to  7  285 

\Valnut    common     .... 

2,830 

Art.  59.] 


TIMBER. 


497 


Table  I.  contains  the  results  of  experiments  on  the  shearing 
of  small  specimens  along  the  fibres,  by  the  late  Mr.  R.  G.  Hat- 
field  ("  Transverse  Strains,"  1877).  5  is  the  ultimate  shearing 
resistance  in  pounds  per  square  inch.  There  were  about  nine 
experiments  for  each  kind  of  timber. 

Table  II.  contains  the  results  of  experiments  by  Mr.  John 
C.  Trautwine  on  round  specimens  0.625  inch  in  diameter,  and 
across  the  fibres  ("Journal  of  the  Franklin  Institute,"  Feb. 

TABLE    III. 

Along  Fibres. 


NO. 

KIND   OF   MATERIAL. 

SHEARING    AREA 
IN   SQUARE 
INCHES. 

ULT.   SHEAR  IN 
POUNDS   PER  SQ. 
INCH. 

DIRECTIONS  TO   RINGS 
OF  GROWTH. 

I 

Oregon  pine  

5.0  and  14.0 

442  and  1,096 

Perpendicular  (2  exps.) 

2 

Oregon  pine  ... 

10.7 

820 

Oblique. 

Oregon  maple 

14.4 

4^6 

Perpendicular. 

Oregon  maple  

10.9 

1,028 

Oblique. 

California  laurel 

ii  .0  and  14.2 

549  and  1,204 

Oblique  (2  exps.) 

6 

Ava  Mexicana  .... 

14.8 

146 

Perpendicular 

i 

9 

10 

ii 

12 
13 

"4 

Je 

Ava  Mexicana    
Oregon  ash  
Oregon  ash      
Mexican  white  mahogany.  .   . 
Mexican  cedar    
Mexican  cedar  .  .    . 
Mexican  mahogany    
Mexican  mahogany  
Oregon  spruce 

11.  0 

14.6 

8.2 

ii.  o  and  15.1 
15-0 
9.8 
15-0 
ii.  i 

700 
443 
1,126 
438  and  1,000 
423 

% 

i<333 
261  and     356 

Parallel. 
Parallel. 
Perpendicular. 
Oblique  (2  exps.) 
Perpendicular. 
Parallel. 
Parallel. 
Perpendicular. 
Parallel  (2  exps  ) 

T6 

Oregon  spruce  

5  5 

315 

Perpend  icu  lar. 

17 

White  pine 

381  and     423 

Perpendicular  (2  exps  ) 

18 

19 
20 

White  pine  
Whitewood  
Whitewood    

16.0    "     24.0 
73    "     ii.  o 
10.8          21.9 

324    "       352 
127            370 
328            481 

Parallel  (2  exps  ) 
Oblique  (2  exps  ) 
Parallel  (2  exps.) 

21 

Whitewood 

322    "       385 

Perpendicular  (2  exps.) 

22 

Yellow  pine  

13.0    "     13.1 

317    "        399 

Oblique  (2  exps.) 

27 

Yellow  pine 

17  o    ''     25.4 

280             409 

Perpendicular  (2  exps.) 

24 

Ash  

16.3    "     24.4 

592              ooo 

Parallel  (2  exps.) 

2S 

Ash                        .          .   . 

16.2    u     16.0 

458    "        700 

Perpendicular  (2  exps.) 

26 

Red  oak 

743              745 

Perpendicular  (2  exps.) 

27 
28 

Red  oak  
White  oak 

16.0    "     24.0 
16.2    "     24  o 

726    u        999 
803    u        966 

Parallel  (2  exps.) 
Parallel  (2  exps  ) 

29 

White  oak  
Yellow  birch 

15.8  "   24.0 

17  o          17  o 

752    "        846 
563              815 

Perpendicular  (2  exps.) 
Oblique  (2  exps.) 

31 

Yellow  birch  
White  maple  

25.6    "     25.6 
16.0 

612              672 
647 

Perpendicular  (2  exps.) 
Oblique. 

White  maple.                   .   . 

15  9  and  24  o 

399  and     537 

Perpendicular  (2  exps.) 

15  8    "    23  8 

235    "        347 

Parallel  (2  exps.) 

Spruce  .... 

16.0    lk    24.0 

316    "        374 

Perpendicular  (2  exps.) 

32 


498  TORSION.  [Art.  60. 

1880).     As  before,   5  is   the    ultimate   shearing   resistance  in 
pounds  per  square  inch. 

Table  III.  has  been  condensed  from  the  results  of  Col. 
Laidley's  tests  at  the  Watertown  Arsenal  (Ex.  Doc.  No.  12, 
47th  Congress,  1st  Session).  Usually,  two  such  results  have 
been  selected  as  will  give  a  correct  idea  of  the  resistance.  In 
all  cases  except  Nos.  19,  20,  23  and  33,  the  smaller  resistance 
belongs  to  the  larger  shearing  surface.  In  No.  33  the  smaller 
resistance  belongs  to  an  unsatisfactory  experiment. 


Art.  60. — Torsion. 
Coefficients  of  Elasticity. 

The  coefficients  of  elasticity  for  torsion  or  shearing  have 
been  given  in  Art.  58,  and  need  not  be  repeated  here. 

Ultimate  Resistance  and  Elastic  Limit. 
WROUGHT  IRON. 

In  1866  Mr.  Kirkaldy  tested  four  hammered  Swedish  iron 
bars  turned  to  a  diameter  of  1.5  inches  for  a  length  of  seven 
diameters.  The  average  ultimate  moment  of  torsion  was  pro- 
duced by  a  weight  of  2,677  pounds  with  a  leverage  of  12 
inches;  hence,  in  Eq.  (83)  of  Art.  10 ;  M  =  2,677  X  12  = 
32,124.  Putting  2rQ  =  d  =  1.5  inches  in  that  equation,  there 
will  result : 

M 

Tm  —  5.1  -^  —  48,540  pounds  per  square  inch. 

This  is  the  greatest  intensity  of  torsional  shear  in  the  sec- 
tion. 


Art.  60.]  WROUGHT  IRON.  499 

If    Tm  be  taken  at  48,000  the  diameter  of  a  wrought-iron 
shaft  required  to  resist  an  ultimate  moment  M,  will  be  : 


d  —  0.047 


If  the  working  moment  be  taken  at  one-eighth  the  ulti- 
mate, then  the  diameter  required  will  be  : 


d  =  0.047  v8  MI  —  0.094  yMt        .      .     .     (2) 

in  which  Mt  is  the  working  moment. 

If  H  is  the  number  of  horse  powers  per  minute  to  be  trans- 
mitted by  the  shafting,  and  n  the  number  of  revolutions  which 
it  is  to  make  : 

H 


Putting  this  value  in  Eq.  (2)  : 

d=  3-742  .......     (4) 


This  value  of  d  will  be  much  too  small  in  the  case  of  long 
shafting  required  in  the  distribution  of  power,  in  consequence 
of  the  bending  caused  by  the  belting. 

The  mean  torsional  moment  at  the  elastic  limit,  in  Mr. 
Kirkaldy's  four  experiments,  was  about  0.4  the  ultimate. 

In  1846  Major  Wade  ("  Experiments  on  Metals  for  Can- 
non ")  tested  three  wrought-iron  circular  cylinders  about  1.9 
inches  in  diameter  and  15  inches  long,  with  the  following  re- 
sults : 


5oo 


TORSION. 


Art.  60.] 


T    = 


=  28,325  Ibs.  per  sq.  in. 


=  27,525 
=  27,800    " 


II        «        « 

it  «  U 


83,650 

Mean  =  27,900  (nearly). 
If  the  mean  be  taken  at  28,000 : 
d  —  0.056 


(5) 


It  is  seen  that  Major  Wade  found  Tm  much  less  than  Kirk- 
aldy's  value  for  Swedish  iron,  and  d  in  Eq.  (5)  is  correspond- 
ingly greater.  If  H  and  n  carry  the  same  signification  as 
before,  and  if  8  is  the  safety  factor  : 


(6) 


In  all  these  results,  the  moments  are  supposed  to  be  given 
in  inch-pounds,  and  the  resulting  values  of  d  are  consequently 
in  inches. 

CAST  IRON. 

Major  Wade  also  made  tests  on  circular  cylinders  of  cast 
iron  about  1.9  inches  in  diameter  and  15  inches  long. 

If  d  is  the  diameter  =  2r0  in  Eq.  (83)  of  Art.  10,  he  found 

the  following  results  with  the  grades  of  iron  shown  : 

• 

2d  fusion Tm  =  31,500  pounds  per  square  inch. 

3d  fusion Tm  —  44.775 

2d  and  3d  fusion Tm  =  49,735 

2d  fusion Ttn  =  40,020 

3d  fusion Ttn  —  53,380 

2d  fusion Tm  —  49, 526 

3d  fusion T,n  —  46,230 

Mean  =  45,000  (nearly). 


Art.  60.]  CAST  IRON,  501 

Hence  the  diameter  in  inches,  for  the  ultimate  moment  M 
in  inch-pounds  is : 


(7) 


These  values  of  Tm  are  very  high,  because  the  iron  with 
which  Major  Wade  experimented  was  evidently  of  a  special 
character  and  extraordinarily  strong. 

The  same  experimenter  tested  some  square  sections,  for 
which,  by  Eq.  (73)  of  Art.  10  : 

M 
T  —  5  —  ;  (b  =  side  of  square)     ....     (8) 


The  following  are  from  Major  Wade's  results  : 

b  =  i.oo  inches  ;  M  —  8,750  inch-pounds  ;  Tm  =  43,750  pounds. 
b  =  1.42  inches  ;  M  =  23,000  inch-pounds  ;  Tm  —  40,210  pounds. 
b  =  1.75  inches  ;  M  =  54,000  inch-pounds  ;  Tm  =  50,370  pounds. 

The  mean  of  these  results  is  :   T  =  44,800  (nearly). 
Hence  for  the  ultimate  moment  in  inch-pounds  : 


b  =  J-  =  0.0481  A*  (9) 

44, 


It  is  to  be  observed  that,  according  to  these  experiments, 
Tm  is  the  same  for  circular  and  square  sections  ;  a  result  very 
different  from  that  of  Prof.  Bauschinger's  experiments,  as  will 
presently  be  seen. 

Four  of  Major  Wade's  experiments  on  hollow  circular  cyl- 
inders are  next  to  be  given. 

Since  Txr  =  o,  in  Eq.  (78)  of  Art.  10,  the  resisting  moment 


502  TORSION.  [Art.  60. 

of  such  a  cylinder,  if  d  is  the  external  and  d^  the  internal  di- 
ameter, will  be  : 


M 


T  /7s  _   T'  d*         T 
=-      -  L  =  ^4~4 


For  the  first  case  : 

d  —  3.25  ins. ;  </x  =  2.61  ins.;  M  =  95,000  in.-lbs. 
Substituting  in  Eq.  (11)  : 

Tm  =  24,170  Ibs.  per  sq.  in.  (nearly). 

For  the  second  case  : 

d  —  2.21  ins. ;  */x  =  1.54  ins. ;  M  =  49,500  in.-lbs. 
Substituting  in  Eq.  (n)  : 

Tm  =  30,610  Ibs.  per  sq.  in.  (nearly). 
For  the  third  case  : 

d  =  i. 8 1  ins.  ;  dI  =  0.91  in.  ;  M  —  37,250  in.-lbs. 
Substituting  in  Eq.  (n)  : 

Tm  =  34,220  Ibs.  per  sq.  in.  (nearly). 
For  the  fourth  case : 

d  =  1.30  ins. ;  4  =  0.65  in. ;  M  =  13,000  in.-lbs. 


Art.  60.]  CAST  IRON.  503 

Substituting  in  Eq.  (11) : 

Tm  —  32,180  Ibs.  per  sq.  in.  (nearly). 

These  results  indicate  that  Tm  decreases  as  the  thickness  of 
the  wall  of  the  hollow  cylinder  decreases  and  as  the  exterior 
diameter  increases. 

Professor  Bauschinger  (Der  Civilingenieur,  1881,  heft  2) 
tested  cylinders  about  40  inches  long,  and  with  the  following 
cross  sections  and  approximate  dimensions : 

Circle Diameter    =      3.25  inches. 

2.30  inches. 


Ellipse Diameters  = 

Square Sides  =   - 


4.40  inches. 
3 .  oo  inches. 
3.00  inches. 
2.04  inches. 


The  ultimate  twisting  moments  substituted  in  Eqs.  (83), 
(41),  (73),  (75),  and  (77)  of  Art.  10,  give  : 

For  Circle  ..................  Tm  =  27,730  pounds  per  square  inch. 

For  Ellipse  .................  Tm  =36,  120  pounds  per  square  inch. 

For  Square  .................  Tm  —  37,i6o  pounds  per  square  inch. 

For  Rectangles  (sides  2  to  i).  .  Tm  =  36,370  pounds  per  square  inch. 
For  Rectangles  (sides  4  to  i).  .  Tm  =  37,090  pounds  per  square  inch. 

These  experiments  give  Tm  considerably  less  value  for  the 
circular  cross  section  than  for  the  others. 

The  U.  S.  Board,  however,  found  the  following  values  for 
four  cast-iron  cylinders  one  inch  long  and  0.565  inch  in  diam- 
eter: 

Tm  =  35>98°;  34>no;  34,280,  and  33,770  Ibs.  per  sq.  in. 


Elas.  Lim.  =  60;  55  ;  64,  and  62  per  cent,  of  Tm,  respect- 
ively. 


504  TORSION.  [Art.  60. 


STEEL. 

In  connection  with  the  torsional  resistance  of  steel,  tests  of 
circular  cylinders  only  are  to  be  considered.  Those  to  first  re- 
ceive attention  were  made  by  Mr.  Kirkaldy  on  English  steel, 
in  1866-1870,  and  the  results  have  been  deduced  from  his  data. 

As  the  sections  are  all  circular,  Eq.  (83)  of  Art.  10  is  the 
only  one  needed  : 


In  this  equation  Tm  is  the  greatest  intensity  of  torsional 
shear,  in  any  section,  in  pounds  per  square  inch;  "  d"  the  di- 
ameter of  the  shaft  or  cylinder  in  inches  ;  and  M  the  twisting 
moment  in  inch-pounds. 

In  all  the  following  experiments  the  lever  arm  of  the  twist- 
ing couple  was  12  inches;  hence,  if  P  is  the  twisting  force, 
M  =  I2P,  and  Eq.  (12)  becomes 


03) 


The  mean  of  four  experiments  with  Bessemer  steel  gave 
for  the  ultimate  resistance 

P  —  2,307  Ibs.,  with  d  =  1.25  inches  ; 

.*.  Tm  —  72,298  Ibs.  per  sq.  in.    .     .     .     .     (14) 

The  length  was  10  inches. 

The  mean  of  some  results  with  Krupp's  cast  steel  in  speci- 
mens 1.25  inches  in  diameter,  and  2.5  inches  for  torsion  length, 
gave: 

P=  2,867  Ibs.     .-.     Tm  =  89,847  Ibs.      .      .      (15) 


Art.  60.] 


STEEL. 


505 


The  following  set  of  results  were  obtained  from  2-inch 
square  bars  turned  down  to  1.382  inches  in  diameter  for  a 
length  of  1 1  inches,  and  gives  the  means  of  the  number  of  tests 
indicated. 


SPECIMENS.                                 /'(ULTIMATE).       7"^  (ULTIMATE).        STRAIN. 

5  Hammered  tires,     ~\  ^ 

3,450  Ibs 

80,006  Ibs 

0.014  " 

5                       axles,      |  ^ 

3,293 

76,365 

O.OII 

4           "          rails,       Vp 

3,248 

75,321 

O.OI2 

4  Rolled  tires,  axles   j  J>  <" 

and  rails.               J 

3,226 

74,811 

0.008 

5  Hammered  tires,      1  o> 

3.562 

82,603 

0.014 

4           "          axles,      I|-B 

3,786 

87,797 

0.013 

ii           •!        i  y  w 

I                       rail,            g£ 

4,054 

94,012 

0.016  | 

i  Rolled  rail.               J  <-> 

3,002 

69,616 

0.012  J 

•    (16) 


The  elastic  strain  is  the  fraction  of  a  complete  turn  made 
by  the  specimen  at  the  elastic  limit. 

The  mean  of  the  Bessemer  steels  in  (14)  and  (16)  give: 

Mean  Tm  =  75,760  Ibs.  per  sq.  in. 

Hence,  if  M  is  the  breaking  moment  of  the  twisting  couple 
in  incli-pounds,  the  following  will  be  the  diameter  of  the  shaft 
in  inches : 


~    

0.0407^;  ....  (17) 


Or,  if  n  is  the  safety  factor,  and  M^  the  greatest  working 
moment : 

d  =  0.0407^*^ (18) 

The  mean  of  the  crucible  steel  results  in  (16),  with  the  ex- 
ception of  the  last,  is: 

Mean  Tm  =  88,140  Ibs. 


506  TORSION.  [Art.  60. 

Hence  the  diameter  (in  inches)  of  the  shaft  which  will  just  sus- 
tain the  breaking  moment  M,  in  inch-pounds^  is: 


d  '=  =  o-o387^^    ....     (19) 


Or,  if  n  is  the  safety  factor,  and  MI  the  greatest  working 
moment : 


d  =  0.0387^^ (20) 

In  all  the  preceding  experiments  the  elastic  limit  varied 
from  40  to  47  per  cent,  of  Tm  (ultimate)  as  given  in  (14),  (15) 
and  (16). 

In  1873  Mr.  Kirkaldy  made  some  experiments  on  specimens 
of  Fagersta  steel  which  possessed  a  length  of  about  9  inches 
and  a  diameter  of  1.128  inches,  the  length  of  the  twisting  lever 
being  still  12  inches.  Eq.  (13)  then  gives  the  following  results, 
each  being  a  mean  of  three  tests : 

MARK.  P  (ULTIMATE).  T?n  (ULTIMATE).  STRAIN. 

1.2 2,120  Ibs 90,397  Ibs 0.29 

0.9 2,336  "  99>6°7  "   °-79 

0.6 2,261  "  96,409  "   i. 02 

0.3 1,520  "  64,813  "   3.22 

The  "strain"  is  the  number  of  complete  turns  made  by  the 
specimen  at  the  place  and  instant  of  rupture. 

The  specimens  with  the  higher  "mark"  numbers  were  the 
higher  steels. 

The  elastic  limit  varied  from  46  to  58  per  cent,  of  the  ulti- 
mate Tm. 

The  diameter  of  a  shaft  for  any  of  these  grades  may  readily 
be  computed  by  the  use  of  these  values  of  Tm  in  equations 
similar  to  Eqs.  (17)  to  (20). 


Art.  60.]  STEEL.  507 

The  following  values  were  determined  by  the  Committee 
on  Chemical  Research  of  the  U.  S.  Board,  "  Ex.  Doc.  23, 
House  of  Rep.,  46th  Congress,  2d  Session,"  with  specimens  I 
inch  long  turned  to  diameters  of  0.625  and  0.565  inch,  and 
tested  in  a  Thurston  machine: 


ELAS.  LIMIT  IN  PER  CENT.  ULT.  ANGLE  OF 

OF    TV.  TORSION. 


100,990  Ibs.  per  sq.  in  

•  ...  34  

149°.  o 

95,230  ,«  «  <<  „  

....  34  

.....142°.  3 

110,260  "  "  "  "  

-..-  33  

68°.  4 

115,780  "  "  "  "  

42  

56°.  I 

52,375  "  " 

34  

278°.  2 

71,420  "  "  "  "  

•.••  45  

220°.  8 

88,210  "  "  "  "  

....  39  

99°-5 

55,885  "  "  "  "  

....  35  

,.i65°.o 

119,040  "  "  "  "  

....  40  

84°.  9 

75,430  "  "  "  "  

44  

i8o°.7 

91,690  "  "  "  "  
96,450  "  "  "  "  
109,010  "  "  "  "  

....  39  
....  36  
••••  29  

53°  to  113°  ' 
48°  to  84° 
61°  to  143° 

Ill 

107,315  "  "  "  "  

....  32  

42°  to  123° 

1*1 

109,590  "  "  "  "  

....  32  

73°  to  141°. 

IV 

Each  of  the  last  five  results  is  a  mean  of  eight  tests. 

The  first  portion  of  these  results  would  possess  more  value 
if  the  test  specimens  had  been  larger. 

With  these  values  of  Tm,  the  diameter  of  a  shaft,  with  the 
torsion  moment  Mm  inch-pounds,  becomes: 


=  1.721 


COPPER,  TIN,  ZINC,  AND  THEIR  ALLOYS. 

The  following  values  of  Tm  have  been  computed  by  the  aid 
of  Eq.  (12)  from  data  determined  by  Prof.  R.  H.  Thurston,  and 
given  by  him  in  the  works  already  cited  in  connection  with 


5o8 


TORSION. 


[Art.  60. 


tension  and  compression.  The  test  specimens  were  0.625  inch 
in  diameter,  with  a  torsion  length  of  i.oo  inch,  and  were  tested 
in  his  torsion  machine.  The  ultimate  shearing  resistances  of 
these  alloys  in  torsion  are  thus  seen  to  vary  as  widely  as  their 
tensile  resistances. 

TABLE   I. 


COMPOSITION. 

• 

ULTIMATE      TORSIVE 

ELASTIC  LIMIT  ;  PER 

ULTIMATE    TORSION 

SHEAR,    Tm. 

CENT.  OF    Tm. 

ANGLE. 

Cu. 

Sn. 

Pounds. 

Degrees. 

100 

OO 

35,9*0 

35 

153-0 

IOO 

oo 

28,430 

40 

52  to   154 

oo 

IOO 

3,196 

45 

557-0 

00 

IOO 

3,297 

33 

691.0 

90 

10 

43,943 

4i 

II4-5 

80 

20 

47,671 

62 

16.3 

70 

30 

4,407 

IOO 

i-5 

62 

38 

i,77o 

IOO 

I.O 

52 

48 

686 

IOO 

I.O 

39 

61 

5,881 

IOO 

1.7 

29 

7i 

5,257 

IOO 

2-34 

IO 

90 

5,76i 

63 

131.8 

90 

10 

25,027 

49 

57-2 

90 

IO 

31,851 

57 

.        72.6 

Tm  is  in  pounds  per  square  inch. 

Table  I.  relates  to  alloys  of  copper  and  tin,  and  Table  II. 
to  alloys  of  copper  and  zinc. 

None  but  specimens  with  circular  sections  were  tested. 

With  the  preceding  values  of  TmJ  the  following  expression 
for  the  diameter  in  inches  may  be  written,  if  M  is  given  in  inch- 
pounds  : 


Art.  60.] 


COPPER-ZINC  ALLOYS. 


509 


TABLE   II. 


PERCENTAGE   OF 

ULTIMATE   TORSIVE 

ELASTIC    LIMIT  ;   PER 

ULT.   TORSION 

SHEAR,     TM. 

CENT.    OF    Tm. 

ANGLE. 

Copper. 

Zinc. 

Pounds. 

Degrees. 

90.56 

9.42 

35,ioo 

17.2 

458.0 

81.90 

17.99 

41,575 

27-5 

345-0 

71.20 

28.54 

41,000 

24.0 

269.0 

60.94 

38.65 

48,520 

29.4 

202.0 

55-15 

44-44 

52,320 

32-7 

lOg.O 

49  66 

50.14 

43,154 

36.0 

38.0 

41.30 

58.12 

4,588 

IOO.O 

1.8 

32.94 

66.23 

7,241 

IOO.O 

1.2 

20.81 

77.63 

16,374 

IOO.O 

0.8 

10.30 

83.88 

\22,5OO 

85.6 

7-i 

o.oo 

IOO.OO 

9,186 

38.1 

i4i-5 

TIMBER. 


In  the  July,  1873,  number  of  Van  Nostrand's  Magazine, 
Prof.  Thurston  gave  the  results  of  some  experiments  on  timber 
test  specimens  of  circular  section,  0.875  inch  in  diameter.  Eq. 
(12)  may  be  written  as  follows  : 


M  =  ^  d*  =  Cd* 


(21) 


Prof.  Thurston  determined  the  values  of  C,  and  the  values 
of  Tm  =  5.  i  C  have  been  computed  from  them  : 

Tm  (per  sq.  in.) 

White  pine 1,530  pounds. 

Yellow  pine,  sap 2,142       " 

Yellow  pine,  heart , 2,448       " 

Spruce  1,836       " 

Ash 2,632       " 

Black  walnut 3,366       " 


510  TORSION.  [Art.  60. 

Tm  (per  sq.  in.). 

Red  cedar 1.958  pounds. 

Spanish  mahogany 3>9?8 

Oak 3,244 

Hickory 5, 202 

Locust 4,896 

Chestnut 2,142 

It  is  presumed  that  the  axis  of  torsion  was  parallel  to  the 
fibres,  which  would  cause  the  shear  to  take  place  across  .the 
latter. 

It  is  interesting  to  observe  that  Tm  is  generally  considerably 
less  than  the  ultimate  resistance  to  simple  shear  as  given  in 
Table  II.  of  Art.  59. 

If  d  is  in  inches  and  M  in  inch-pounds,  there  may  again  be 
written  : 


M 


If  M  is  given  in  foot-pounds,  12M  is  to  be  written  for  M. 
If  J/x  is  the  greatest  working  moment,  and  n  the  safety  factor, 
nMI  is  to  be  written  for  M. 


Relation  between  Ultimate  Resistances  to  Tension  and  Torsion. 

In  the  "  Trans.  Am.  Soc.  of  Civ.  Engrs.,"  Vol.  VII.,  1878, 
Prof.  Thurston  gave  the  results  of  some  of  his  experiments 
which  were  made  with  a  view  to  the  determination  of  this  re- 
lation. If  Mis  the  ultimate  torsional  moment  in  foot-pounds 
of  specimens  one  inch  long  and  0.625  inch  in  diameter  ;  6  the 
angle  of  torsion  corresponding  to  this  greatest  moment  M\ 
and  T  the  ultimate  tensile  resistance  in  pounds  per  square 
inch  ;  he  deduced  from  a  large  number  of  steel  specimens  of 
wide  range  in  grades  the  following  formula : 


Art.  60.]  TENSION  AND    TORSION.  5  1  1 


No  experiments  were  made   in  which  6  was  greater  than 

°. 

T  is  thus  seen  to  increase  as  M  increases  and  as  6  decreases. 


CHAPTER    IX. 
BENDING  OR  FLEXURE. 

Art.  61.— Coefficient  of  Elasticity. 

The  coefficient  of  elasticity,  as  determined  by  experiments 
in  flexure,  can  scarcely  be  considered  other  than  a  conven- 
tional quantity.  If  the  coefficients  of  elasticity  for  pure  ten- 
sion and  compression  were  exactly  equal  to  each  other,  and  if 
all  the  hypotheses  involved  in  the  common  theory  of  flexure 
were  true,  then,  indeed,  the  coefficient  of  elasticity  for  flexure 
would  possess  actual  existence,  and  be  the  same  as  that  for 
either  tension  or  compression. 

These  conditions,  however,  never  exist,  and  the  quantities 
found  in  this  chapter  under  the  name  "  coefficient  of  elasticity  " 
possess  value  chiefly  as  empirical  factors  which  enable  the 
deflections  in  the  different  cases  to  be  estimated  with  sufficient 
accuracy  for  all  ordinary  purposes. 

The  formulae  to  be  used  in  the  determination  of  the  co- 
efficients of  elasticity  for  flexure  have  already  been  established, 
and  their  use  will  be  shown  in  succeeding  Articles. 

Art.  62. — Formulae  for  Rupture. 

As  with  the  formulae  for  the  coefficient  of  elasticity,  so  with 
the  formulae  for  rupture  in  bending  ;  they  are  all  deductions 
from  the  common  theory  of  flexure,  and,  strictly  speaking,  are 
subject  to  all  the  limitations  involved  in  it. 

If  K  and  K'  are  the  greatest  intensities  of  stress  in  the  sec- 


THE 


Art.  62.]  FORMULA  FOR  RUPTURE. 

tion  of  rupture  and  at  the  instant  of  rupture  ;  y  the  variable 
normal  distance  of  any  fibre  from  the  neutral  surface  ;  y^  and 
y  the  greatest  values  of  y ;  b  the  variable  width  of  the  section 
(normal  toj)  ;  and  M  the  resisting  moment  at  the  instant,  of 
rupture  ;  then  the  general  formula  for  rupture  by  bending,  as 
given  by  Eq.  (i)  of  Art.  27,  is: 


fbdy.     .     .     .     (i) 

y    -y 

This  equation  is  based  on  the  supposition  that  the  coeffi- 
cients of  elasticity  for  tension  and  compression  are  not  equal. 
Although  this  supposition  is  strictly  true,  yet  equality  is  al- 
most invariably  assumed  ;  particularly  in  the  treatment  of 
solid  beams.  Fortunately,  this  assumption  is  not  far  wrong  in 
those  materials  which  are  most  valuable  to  the  engineer. 

Eq.  (  i  ),  however,  will  hereafter  be  applied  to  some  cast-iron 
flanged  beams. 

If  the  tensile  and  compressive  coefficients  of  elasticity  are 

fC        fC' 
equal,  --  =  —  r  .     Or,  if  K  is  the  greatest  intensity  of  stress  in 

the  section  which  exists  in  the  fibre  at   the  greatest  normal 

K       K 
distance,  d»  from  the  neutral  surface,  then  --  =  -j,and  Eq.  (i) 

becomes  : 


(2) 


This  is  Eq.  (14)  of  Art.  18,  and  is  the  one  almost  invariably 
used  in  engineering  practice. 

In  Eq.  (2)  I  is  the  moment  of  inertia  of  the  cross  section  of 
the  beam  about  its  neutral  axis.  By  introducing  the  value  of 
/  for  each  particular  shape  of  section,  simple  working  forms  of 
Eq.  (2)  may  easily  be  obtained.  This  will  be  done  for  two 
sections  in  the  following  Article. 
33 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


Art.  63. — Solid  Rectangular  and  Circular  Beams. 

While  the  rectangular  form  of  cross  section  almost  invari- 
ably characterizes  timber  beams,  similar  ones  of  iron,  steel  and 
other  metals  are  only  occasionally  seen.  Beams  of  iron  and 
steel  with  circular  cross  sections,  however,  are  quite  common 
as  pins  in  pin  connection  bridges. 

If  2Px  represents  the  moment  of  the  external  forces  about 
the  neutral  axis  of  any  section,  Eq.  (2)  of  the  preceding  Article 
becomes : 

KI 
=  -r (i) 


The  following  are  the  values  of  /  and  d*  for  rectangular  and 
circular  sections,  h  being  the  side  of  the  rectangle  normal,  and 
b  that  parallel  to  the  neutral  axis,  while  r  is  the  radius  of  the 
circular  section,  and  A  the  area  in  each  case  : 


Rectangular:  •< 


bfc 

12 

h 

2 


Ah* 

12 


Circular : 


Ar* 


If  the  beams  are  supported  at  each  end  and  loaded  by  a 
weight  Wat  the  centre  of  the  span  (or  distance  between  sup- 
ports), which  may  be  represented  by  /,  then  the  moment  at  the 
centre  of  the  beam  becomes : 


Art.  63.  ]  WRO  UGHT  IRON.  5  1  5 

Wl 
^Px  =  M  =  —     ......     (2) 

4 

There  will  then  result  from  Eq.  (i)  : 
For  rectangular  sections  : 


466 

For  circular  sections  : 

Wl       nKr*       KAr 

M  =  -  =  --  =  -     .....    (4) 
444 

The  quantity  K  is  called  the  modulus  of  rupture  for  bending, 
and  if  experiments  have  been  made,  so  that  Wis  known,  Eq. 
(3)  gives  : 

3^Z       _3^7 
2  Ah  ~~  2  b& 

And  Eq.  (4)  : 

Wl       Wl 


If  the  rectangular  section  is  square,  bh*  =  b*  =  hz. 


Wrought  Iron. 

If  the  beam  is  simply  supported  at  each  end  and  carries  a 
load  £Fat  the  centre,  while  E  is  the  coefficient  of  elasticity  and 
w  the  deflection  at  the  centre,  Eq.  (8)  of  Art.  24  gives : 

Wl* 


FLEXURE   OF  SOLID  BEAMS.  [Art.  63. 

If,  in  any  given  experiment,  w  is  measured,  E  may  then  be 
found  by  the  following  form  of  Eq.  (7)  : 


(8) 

O'WJ. 

If  the  section  is  rectangular : 

Wfi 

-    .    (9) 


Mr.  Edwin  Clark  tested  a  one-inch  square  wrought-iron  bar 
with  the  following  results  at  the  "  elastic  limit  :  " 

/  —  12  inches.  W  =  2,636.00  Ibs. 

w  =  0.09  inch.  b  =  h  —  I  inch. 

Eq.  (9)  then  gives  : 

E  =  12,652,809.00  pounds  per  square  inch. 

The  mean  for  2  one  and  a  half  inches  square  bars  was  as 
follows : 

/  =  36  inches.  W  =  2,766.00  Ibs. 

w  =  0.305  inch.  b  =  h  —  1.5  inches. 

.'.     E  =  20,894,600.00  pounds  per  square  inch. 

A  mean  of  4  two  inches  square  bars  of  Swedish  iron,  tested 
by  Mr.  Kirkaldy,  in  1866,  gave  the  following  results  at  the 
"  elastic  limit :  " 

/  =  25  inches.  W  —  6,625.00  Ibs. 

w  =  0.082  inch.  b  =  h  =  2  inches. 

E  —  19,725,000.00  pounds  per  square  inch. 


Art.  63.]  WROUGHT  IRON.  5 1/ 

By  "  weighting"  these  results  in  proportion  to  the  number 
of  tests  of  which  each  is  a  mean,  the  mean  of  all  becomes : 

E  =  19,049,000.00  pounds  per  square  inch. 

It  is  very  probable  that  if  w  had  been  measured  for  smaller 
loads,  E  would  have  been  materially  increased. 

Mr.  Kirkaldy  tested  the  same  four  square  Swedish  iron  bars 
to  rupture.  By  the  aid  of  Eq.  (5),  and  the  data  given  above, 
the  greatest,  mean,  and  least  results  were  as  follows : 


W.  K.  FINAL  DEFLECTION. 

Greatest 15,885  Ibs 74,475  Ibs.  per  sq.  in 5.85  ins. 

Mean 14,516  Ibs 68,044  Ibs.  per  sq.  in 5.35  ins. 

Least 13,338  Ibs 62,522  Ibs.  per  sq.  in 4.98  ins. 


The  ultimate  tensile  resistance  of  the  same  iron  was  about 
45,000  pounds  per  square  inch.  These  experiments  would  seem 
to  show  that  K,  for  square  bars  under  similar  circumstances 
of  span  and  depth,  may  be  taken  about  1.5  times  the  ultimate 
resistance  to  tension. 

The  results  in  the  following  table  were  computed  by  the  aid 
of  Eq.  (6),  for  some  circular  beams  of  "  Burden's  Best "  iron, 
which  were  tested  at  the  Rensselaer  Polytechnic  Institute  in 
November,  1882.  As  beams  cannot  be  actually  broken  under 
such  circumstances,  the  "  ultimate  "  value  of  K  was  taken  with 
a  final  deflection  of  one  to  one'and  quarter  the  diameter. 

The  "  elastic  limit  "  is  taken  at  that  point  beyond  which  the 
metal  "  flows,"  and  is  indicated  by  the  incapability  of  the  spec- 
imen to  hold  up  the  scale  beam  beyond  it,  under  a  small  in- 
crease of  stress ;  in  other  words,  it  is  that  point  at  which  the 
specimen  "  breaks  down." 

These  experiments  show  conclusively  that  "ultimate"  K 
decreases  as  the  ratio  of  span  over  diameter  increases,  but  they 


5i8 


FLEXURE  OF  SOLID  BEAMS. 


[Art.  63. 


Circular  Beams  of  "  Burden* s  Best "  Wrought  Iron. 


I 

V. 

^ 

r 

KIND. 

DIAMETER. 

SPAN. 

Elastic. 

Ultimate. 

Elastic. 

Ultimate. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Turned  . 

1.25 

12 

3,000 

6.OOO 

46,950 

93,900 

Turned  . 

1.25 

8 

4,400 

10,500 

45,900 

109,500 

Turned  . 

•25 

12 





54,760 

93,870 

Turned  . 

•25 

"    8 





52,150 

114,700 

Rough.  . 

.00 

12 





55,ooo 

91,700 

Rough.  . 

.00 

8 





57,000 

101,900 

Turned  . 

.00 

12 





55,000 

91,600 

Turned  . 

.00 

8 





107,000 

Rough.  . 

.00 

12 

1,700 

3,OOO 

5i,95o 

91,680 

Rough.  . 

.00 

8 

2,8oo 

4,800 

57,000 

97,800 

Turned  . 

0.75 

12 

700 

1,100 

47,100 

74,050 

Turned  . 

0.75 

8 

1,200 

1,900 

53,88o 

85,310 

Turned.. 

0.75 

12 

700 

1,100 

47,100 

74,050 

Turned  . 

0-75 

8 

I,3OO 

1,900 

58,370    . 

85,3io 

are  not  sufficiently  extended  to  establish  the  limits  of  applica- 
tion of  the  observation. 


Cast  Iron. 

All  the  following  results  for  cast-iron  beams  are  found  from 
Major  Wade's  experiments  ("  Strength  and  other  Properties  of 
Metals  for  Cannon,"  1856).  His  test  bars  were  either  two 
inches  square  in  section  or  two  inches  in  diameter,  and  were 
twenty-four  inches  long.  They  were  loaded  at  the  centre,  and 
the  distance  between  supports  was  twenty  inches.  The  follow- 
ing table  gives  results  for  square  bars.  K  is  given  in  pounds 
per  square  inch,  and  is  found  by  the  aid  of  Eqs.  (5)  and  (6). 
"  Def"  is  the  final  deflection. 

Although  Major  Wade  made  many  other  experiments  of 
the  same  kind,  these  may  be  considered  representative  ones. 


Art.  63.] 


CAST  IRON. 


519 


Bars  with  square  section,  Ea,.  (5). 


. 

HOURS 

KIND  OF  IRON. 

IN 

w. 

K. 

DBF. 

FUSION. 

Lbs. 

Lbs. 

In. 

r 

o 

H,587 

42,130 

0.156 

I 

12,487 

45,  no 

0.152 

2 

15,019 

52,870 

0.152 

2 

15,525 

55,930 

0.147 

, 

i 

11,812 

42,760 

0.162 

Stockbridge  iron  *  2d  lusion                 J 

I 

14,512 

52,670 

0.195 

I* 
2 

16,481 
19,462 

60,  500 
69,680 

0.202 
0.230 

, 

0 

12,987 

49,070 

O.25O 

0 

13,365 

50,120 

O.2I7 

I 

15,363 

57,330 

0.220 

Franklin  iron  *  3d  fusion  •  .    • 

I 

14,616 

54,550 

0.195 

2 

13,788 

48,730 

0.152 

2 

14,850 

50,720 

0.170 

3 

16,056 

56,050 

0.175 

, 

3 

16,722 

60,410 

0.170 

Bars  with  circitlar  section^  Eq.  (6). 


Franklin  iron  ;  3d  fusion  

1 
I* 

10,437 
8,665 

70,600 

57,720 

0.237 
0.166 

Franklin  iron  *  2d  fusion                   . 

31 

i 

2 

II,  112 
10,606 
7,920 
9,270 

70,740 
71,740 
52,360 
63,670 

0.254 
0.24O 

•    0.240 

3 
4 

9,481 
7,920 

64,820 
52,360 

0.262 

It  is  both  interesting  and  important  to  observe  that  K  and 
the  final  deflection  are  materially  larger  for  circular  beams  than 
for  square  ones. 

By  comparing  these  values  of  K  with  the  ultimate  tensile 
resistances  found  by  Major  Wade,  and  which  have  been  given 
under  the  head  of  "  Tension,"  it  will  be  seen  that  no  great 
error  will  be  involved  if  K  is  taken  at  twice  the  ultimate  tensile 


52C 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


resistance  for  square  bars,  and  two  and  a  quarter  times  the  same 
quantity  for  bars  with  'circular  section. 

Whether  these  ratios  will  hold  for  iron  of  inferior  quality  to 
that  used  by  Major  Wade,  can  only  be  determined  by  farther 
experimenting. 


Steel. 


Some  circular  Bessemer  steel  beams  with  12  and  8-inch 
spans  were  tested  at  the  Rensselaer  Polytechnic  Institute  in 
Nov.,  1882,  with  the  results  which  are  given  in  the  next  table. 
The  "  elastic  limit "  is  that  point  at  which  the  specimen 
"  breaks  down*."  The  "  ultimate  "  value  was  that  for  which 
the  deflection  was  equal  to  one  or  one  and  a  quarter  the 
diameter. 

Circular  Bessemer  steel  beams,  Eq.  (6). 


V 

y. 

1 

K 

Elastic. 

Ultimate. 

Elastic. 

Ultimate. 

Turned 

In. 

I    OO 

Ins. 
12 

Lbs. 

Lbs. 

Lbs. 
86,000 

Lbs. 
146,7^0 

I    OO 

8 

8^,300 

152,800 

1  .00 

12 

2  ^OO 

4  "?OO 

76,400 

137,520 

I    OO 

8 

37CO 

7  ^OO 

76  4OO 

152,800 

O    7^ 

12 

I    I5O 

I  8OO 

77  400 

I22,2OO 

O.  7< 

8 

I   8OO 

a  OQO 

80  8OO 

148,200 

O.  7^ 

12 

i  mo 

I   TOO 

77  4.OO 

114,400 

O    7^ 

8 

i  800 

3"  OO 

80  8OO 

148  2OO 

The  "  ultimate  "  K  is  seen  to  decrease  as  the  ratio  of  length 
over  diameter  increases. 

The  following  table  contains  results  computed  from  the  ex- 
periments of  the  "  Steel  Committee  "  of  the  British  Institu- 
tion of  Civil  Engineers;  the  experiments  were  made  in  1868. 


Art.  63.] 


STEEL. 


$21 


The  bars  were  1.9  inches  square  in  section,  and  the  distance 
between  supports  was  twenty  inches. 


JBessemer  Steel,  Eq.  (5). 


ELASTIC   OVER 

FINAL  DEFLEC- 

KIND AND    NUMBER    OF   TESTS. 

K. 

ULTIMATE. 

TION   IN  INCHES. 

Lbs.  per  sq.  in. 

5    tires  hammered  

I2Q  O^O 

O  57^ 

T   82 

5,  axles,         "          

I2Q.325 

O.6l5 

4..O8 

4   rails           "    •          .            

I2C  QOO 

o  612 

q  QA 

4.  tires,  axles,  rails  :  rolled  .  . 

115.  120 

0.561 

4.  01 

Crucible  Steel,  Eq.  (5). 


5    tires,  hammered             . 

14^  5^O 

O  574 

<J     <JO 

4,  axles,        "          . 

ic2  055 

O   C7Q 

•2    OIT 

I,  rail,           "          

17^,470 

o  4^6 

•7    65 

I    axle    rolled     . 

118  1  60 

o  5^8 

<J    84 

Each  result  is  an  average  of  the  number  of  tests  shown  in 
the  left  column. 

The  ratio  "  elastic  over  ultimate  "  is  the  value  of  K  at  the 
"  elastic  limit  "  divided  by  its  ultimate  value  as  given  in  the 
table. 

Table  IX.  of  Art.  34  gives  the  ultimate  tensile  resistances 
of  these  same  steels.  That  table,  taken  in  connection  with  the 
results  just  given,  shows  that  K  is  about  1.66  times  the  ultimate 
tensile  resistance  for  square  Bessemer  steel  bars,  and  about  1.85 
times  the  same  quantity  for  square  crucible  steel  bars. 

Mr.  J.  W.  Cloud,  of  the  Penn.  R.  R.  Co.,  made  bending 
tests  of  the  Bessemer  rail  steel  whose  ultimate  tensile  resist- 
ances are  given  in  Table  IV.  of  Art.  34.  His  test  pieces  were 
12  inches  long,  1.5  inches  wide,  and  0.5  inch  thick.  The  load 
was  applied  in  the  direction  of  the  thickness,  and  midway  be- 
tween supports  10  inches  apart.  The  greatest,  mean  and  least 


522 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


results  of  the  18  means  of  the  groups  shown  in  Table  IV.,  Art. 
34,  are  the  following : 

W.  K. 

Greatest 3,349  Ibs 133,960  Ibs.  per  sq.  in . 

Mean 3,026  Ibs 121,040  Ibs.  per  sq.  in. 

Least 2,765  Ibs 110,600  Ibs.  per  sq.  in. 

With  these  rectangular  specimens  of  Bessemer  rail  steel, 
supported  flatwise,  therefore,  K  may  be  taken  about  1.6  the 
ultimate  tensile  resistance. 

The  following  table  contains  the  results  of  Mr.  Kirkaldy's 
experiments  on  square  bars  of  Fagersta  steel.  These  bars 
were  1.9  inches  square  in  section,  and  rested  on  supports  20 
inches  apart.  W  is  the  breaking  weight  at  centre,  and  K  is 

Fagersta  Steel  Square  Bars. 


MARK. 

W, 

POUNDS. 

K, 

LBS.   PER  SQ.    IN. 

ELASTIC    OVER   ULTI- 
MATE. 

FINAL  DE- 
FLECTION. 

I    2 
1.2 
1.2 

0.9 
0-9 
O.g 

30,496)  [!* 
32,396^  §  « 

35,376  S  ^ 

133,380)      "g 
143,880^     |    | 
154,710)    |    2" 

0.669)      "0. 

0.695  [•   g  ^o 
0.616)  ^  o- 

Ins. 
o  75 

O.72 
0.87 

43,820)    JJs 
44,552  f   |» 
43,128)  |  % 

191,640)      "£ 
I94,850^     |   J 
188,640)     g   ^ 

0.500)    »    • 
0.476      g^ 
0.512)  »g  o 

I.46 
1.62 

I.38 

0.6 
0.6 
0.6 

40,260)    "  i2 

36,  200  1   g  * 
38,120)  g« 

176,100)      "    & 
'    158,310  [     |    g 
166,740)    j^O 

°'467)    ^ 
0.491^  | 
0.482)  |o 

3-15 
3-56 
3-22 

0-3 
0.3 
0-3 

24,42o)    tig 
23,280  [   g  w 
28,150)   jgj? 

I06,800)     "c§ 

101,820  V    g  £ 

1      fli   O 

123,120)      jg    M 

0.561)     "    ^ 

o.653[   |5 
0.654)  go 

5.22 

5-05 
5-05 

Art.  63.]  COMBINED   STEEL  AND  IRON.  523 

computed  by  the  aid  of  Eq.  (5).  The  column  "Elastic  over 
ultimate"  contains  the  ratios  of  the  values  of  K  at  the  "  elastic 
limit "  divided  by  the  ultimate  values  given  in  the  table. 

The  "Mark"  shows  the  character  of  the  steel;  1.2  is  the 
hardest,  and  0.3  the  softest. 

K  is  about  1.6  times  the  ultimate  tensile  resistance  for  the 
grades  1.2  and  0.6,  and  1.8  times  the  same  quantity  for  the 
grades  0.9  and  0.3. 

Combined  Steel  and  Iron. 

In  Sept.,  1 88 1,  some  interesting  and  valuable  experiments 
on  the  transverse  resistance  of  pins  (solid  circular  beams)  were 
made  at  Phcenixville,  Pa.,  by  the  Phoenix  Iron  Co. 

The  pins  were  of  combined  iron  and  steel,  the  core  of  the 
pin  being  of  steel,  and  the  outside  of  iron.  In  such  a  pin  the 
iron  seems  to  change  gradually  to  the  steel,  but  the  shell  of 
iron  may  perhaps  be  considered  one  quarter  to  one  half  an  inch 
thick. 

These  pins  are  supported  at  each  end  and  loaded  in  the 
centre.  The  results  of  the  experiments  are  given  in  the  fol- 
lowing table : 

D    =  diameter  of  pin. 

/      =  length  in  inches  between  supports. 

W  =  weight  (pounds)  at  centre. 

K'  =  intensity  of  stress  per  sq.  in.  on  extreme  fibre,  in 

general. 
K    =  intensity  of  stress  per  sq.  in.  on  extreme  fibre,  at 

rupture. 

K  is  the  greatest  value  of  K'  for  any  one  pin.  Either  K  or 
K',  by  Eq.  (6),  has  the  value : 

Wl 

KorK'  =  %!. 

Ar 


524 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


PIN. 

Z>. 

/. 

w. 

K'  OR  K. 

REMARKS. 

Ins. 

I 

4fV 

24 

26,000 

32,815. 



I 

4iV 

24 

60,000 

54,692. 

Elastic  limit. 

I 

4fV 

24 

100,000 

9M54 

Not  broken.     Deflection  =  f  ins. 

2 

4 

24 

60,000 

40,241. 



2 

42 

24 

84,000 

56,337. 

Elastic  limit. 

2 

4? 

24 

148,000 

99,260.  =  K 

Broken. 

3 

4j 

20.5 

68,000 

38,955. 



3 

4* 

20.5 

84,000 

48,121. 

Slight  permanent  set. 

3 

3 

20.5 

252,000 

144,360.  =  K 

Broken. 

The  mean  of  the  two  values  of  K  is  : 

,.      90,260  4-  144,360 
K  =  *•?.  _—  121,810.00  pounds. 


Copper,  Tin,  Zinc,  and  their  Alloys. 

In  the  following  table  are  given  the  data  and  the  results  of 
the  experiments  of  Prof.  R.  H.  Thurston,  as  contained  in  his 
various  papers,  to  which  reference  has  already  been  made. 
The  distance  between  the  points  of  support  was  twenty-two 
inches,  while  the  bars  were  about  one  inch  square  in  section, 
and  of  cast  metal. 

The  modulus  of  rupture,  K,  is  found  by  Eq.  (5),  in  which, 
however,  in  many  of  these  cases,  W  is  the  weight  applied  at 
the  centre,  added  to  half  the  weight  of  the  bar.  When  K  is 
large  and  the  specimens  small,  this  correction  for  the  weight  of 
the  bar  is  unnecessary  ;  otherwise,  it  is  advisable  to  introduce  it. 

The  coefficient  of  elasticity,  E,  is  found  by  Eq.  (9),  in  which 
Wis  the  centre  load  added  to  five-eighths  of  the  weight  of  the 
bar. 

The  manner  in  which  both  these  corrections  arise,  is  com- 
pletely shown  in  Case  2  of  Art.  24. 


Art.  63.] 


COPPER,    TIN,    ZINC  AND  ALLOYS. 


525 


E,  for  any  particular  bar,  has  a  varying  value  for  different 
degrees  of  stress  and  strain.  Those  given  in  the  table  may 
be  considered  average  values  within  the  elastic  limit. 

As  usual,  "  elastic  over  ultimate "  is  the  ratio  of  K  at  the 
elastic  limit  over  its  ultimate  value. 

An  examination  of  the  ultimate  tensile  and  compressive 
resistances  of  these  same  alloys,  as  given  in  preceding  pages, 
shows  that  the  ratio  of  K  over  either  of  those  resistances  is 
very  variable.  It  is  usually  found  between  them,  but  occasion- 
ally it  exceeds  both. 

Square  Bars. 


PERCENTAGE  OF 

*; 

LBS.  PER  SQ.  IN. 

ELASTIC    OVER 
ULTIMATE. 

FINAL  DEFLEC- 
TION. 

E. 

LBS.  PER  SQ.  IN. 

Cu. 

Sn. 

Zn. 

Ins. 

IOO 

O.OO 

o.oo 

29,850 



8.00 

9,OOO,OOO 

100 

0.00 

0.00 

25,920 

5     0.14 

]  to  0.41 

1.38 

to  8.00 

i    10,830,600 

IOO 

o.oo 

0.00 

21,251 

0.346 

2.31 

I3,986,6OO 

IOO 

0.00 

o.oo 

29,848 

0.140 

Bent. 

10,203,200 

90 

10.00 

0.00 

49,400 

0.400 

Bent. 

14,012,135 

QO 

JO.  OO 

O.  OO 

ef)    -57C; 

o.  41 

n      nf) 

yu 

80 

20.00 

o.oo 

0^,  J/  D 

56,715 

0.657 

J  •  J*» 
0.492 

13,304,200 

70 

30.00 

0.00 

12,076 

I.OO 

O.O62 

15,321,740 

6l.7 

38.3 

o.oo 

2,761 

1.  00 

0.032 

9,663,990 

48.0 

52.0 

o.oo 

3,600 

I.OO 

0.019 

17,039,130 

39-2 

60.8 

o.oo 

8,400 

I.OO 

O.O6O 

12,302,350 

28.7 

71.3 

0.00 

8,067 

0.583 

O.  121 

9,982,832 

9-7 

90-3 

o.oo 

5,305 

0.25 

Bent. 

7,665,988 

o.oo 

IOO 

o.oo 

3,740 

0.273 

Bent. 

6,734,840 

o.oo 

IOO 

0.00 

4,559 

0.267 

Bent. 

5,635,59° 

80.00 

o.oo 

20.00 

21,193 



3-27 

II,OOO,OOO 

62.50 

o.oo 

37.50 

43,216 



3-^3 

14,000,000 

58.22 

2.30 

39-48 

95,620 



1.99 

11,000,000 

55-oo 

0.50 

44-50 

72,308 





92-32 

o.oo 

7.68 

21,784 

0.30 

Bent. 

13,842,720 

82.93 

0.00 

16  98 

23.197 

0.41 

Bent. 

14,425,150 

71.20 

o.oo 

28.54 

24,468 

0.51 

Bent. 

14,035,330 

63-44 

0.00 

36.36 

43,216 

0.53 

Bent. 

14,101,300 

58.49 

o.oo 

41.10 

63,304 

0.48 

Bent. 

11,850,000 

54-86 

o.oo 

44.78 

47,955 

0.39 

Bent. 

10,816,050 

526 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


Square  Bars. —  Continued. 


PERCENTAGE  OF 

A', 

LBS.  PER  SQ.  IN. 

ELASTIC  OVER 
ULTIMATE. 

FINAL  DEFLEC- 
TION. 

4, 

LBS.  PER  SQ.  IN. 

Cu. 

Sn. 

Zn. 

Ins. 

43.36 

0.00 

56.22 

17,691 

1.  00 

0.0982 

12,918,210 

36.62 

0.00 

62.78 

4,893 

1.  00 

0.0245 

14,121,780 

29.2O 

o.oo 

70.17 

16,579 

1.  00 

0.0449 

14,748,170 

20.81 

o.oo 

77.63 

22,972 

I.OO 

0.1254 

14,469,650 

10.30 

o.oo 

88.88 

41,347 

0.73 

0.5456 

12,809,470 

0.00 

0.00 

100.00 

7,539 

0.57 

0.1244 

6,984,644 

70.22 

8.90 

20.68 

50,541 

•  

0.4019 

I4,4OO,OOO 

56.88 

21.35 

21-39 

2,752 



0.0146 

14,800,000 

45-oo 

23-75 

31-25 

6,512 



0.0150 

7,000,000* 

66.25 

23-75 

IO.OO 

8,344 



O.OI62 

12,000,000* 

10.00 

50.00 

40.00 

21,525 



Bent.  ' 

9,000,000 

58.22 

2.30 

39-48 

95,623 



2.000 

10,600,000 

60.00 

10.00 

30  oo 

24,700 



0.1267 

14,500,000 

65.00 

20.00 

15.00 

n,932 



0.0514 

17,000,000 

70.00 

IO.OO 

20.00 

36,520 



0.1837 

15,000,000 

75.00 

5-00 

20.00 

55,355 



Bent. 

13,000,000 

80.00 

IO.OO 

IO.OO 

67,117 



Bent. 

13,500,000 

55-oo 

5.00 

44-50 

72,308 



Bent. 

11,000,000 

60.00 

2.50 

37.50 

69,508 



1.500 

13,000,000 

72.52 

7.50 

20.00 

51,839 



Bent. 

12,000,000 

77-50 

12.50 

10.00 

61,705 



0.705 

13,500,000 

85.00 

12.50 

2-5 

62,405 



Bent. 

12,500,000 

*  These  bars  were  about  half  the  length  of  the  others. 


Timber  Beams. 

As  timber  beams  are  always  rectangular  in  section,  Eq.  (3) 
only  will  be  needed.  Retaining  the  notation  of  that  equa- 
tion, if  the  beam  carries  a  single  weight  W  zk  the  centre  of  the 
span  /: 


3     / 


(10) 


If  the  total  load  W  is  uniformly  distributed  over  the  span  : 


Art.  63.]  TIMBER.  527 


3 


As  K  is  supposed  to  be  expressed  in  pounds  per  square 
inch,  all  dimensions  in  Eqs*.  (10)  and  (n)  must  be  expressed  in 
inches. 

In  the  use  of  timber  beams  it  is  usually  convenient  to  take 
the  span  /  in  feet,  and  the  breadth  (U)  and  depth  (//)  in  inches. 
Placing  12/for  /,  therefore,  in  Eqs.  (10)  and  (n); 

TT,      KAh  KAh 

W=-TsT  '    and'    w   =2TsT    '    '    '    (I2> 

in  which  formulae  /  must  be  taken  in    feet  and  A  and  h  in 
inches. 

J£ 

If  B  be  put  for  —  ,  Eq.  12  becomes  : 

Io 


W=B;     and,      W  =  2B  .     .     .     (13) 

/  / 

Hence,  when  W  and  W  have  been  determined  by  experi- 
ment : 

For  single  load  W  at  centre  : 

''-5  ........  <•* 

For  total  load  W  uniformly  distributed  : 


If  the  beam  has  a  section  one  inch  square  and  is  one  foot 

W 
long,  B  =  W  —  ---  .     By   therefore,   may   be   considered   the 

unit  of  transverse  rupture  ;  it  is  sometimes  called  the  coefficient 
for  centre  breaking  loads. 


528 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


Table  I.  is  a  condensed  statement  of  the  result  of  experi- 
ments by  the  late  R.  G.  Hatfield,  a  complete  account  of  which 
may  be  found  in  his  "  Transverse  Strains,"  1877.  All  the  test 


TABLE   I. 


MATERIAL. 

B. 

K  =  iBS. 

MATERIAL. 

8. 

K  =  i8A 

Georgia  Pine  

Lbs. 

8<;o 

Lbs. 

15,300 

Ash  

Lbs. 

QOO 

Lbs. 
16  200 

I,2OO 

2I,6oo 

Maple  

I   IOO 

19  800 

White  Oak  

6^O 

II  700 

Hickory  

I  O^O 

1  8  900 

Spruce      ...              . 

ceo 

Q  QOO 

Cherry  .  . 

650 

ii  700 

White  Pine 

5OO 

Q  OOO 

Black  Walnut 

7^O 

T-I  coo 

Hemlock 

AK.Q 

8   1OO 

CQO 

10  600 

Whitewood  

6OO 

10  800 

New  England.  Fir  .  . 

oyu 
070 

6  610 

Chestnut            

4.8O 

8  640 

specimens  were  of  American  woods  with  cross  dimensions  va- 
rying from  one  to  two  inches  and  span  of  1.6  feet. 

Table  II.  contains  the  results  of  experiments  on  specimens 
of  American  timber,  given  by  Prof.  R.  H.  Thurston  in  the 
"Journal  of  the  Franklin  Institute,"  Oct.,  1879.  Tne  test 
specimens  were  3  inches  square  and  4.5  feet  between  supports. 
The  coefficient  o.f  elasticity  is  in  pounds  per  square  inch,  and 
is  found  by  Eq.  (9). 

Later  experiments  by  Prof.  Thurston  ("  Jour.  Frank.  Inst.," 
Sept.,  1880),  on  a  great  variety  of  yellow  pine  specimens,  both 
in  respect  to  dimensions  and  degree  of  seasoning,  induced  him 
to  draw  the  following  conclusions  in  regard  to  that  timber : 

The  elasticity  of  yellow  pine  timber  as  used  in  construction 
is  very  variable,  the  coefficient  varying  from  one  to  three  mill- 
ions, the  average  being  about  two  millions  for  small  sections, 
and  a  little  above  one  and  a  half  millions  of  large  timber. 


UNIVERSIT 


Art.  63.] 


TIMBER. 


TABLE  II. 

Specimens  3  ins.  x  3  ins.  x  4-S//. 


ELASTIC 

DEFLECTION 

IN   INCHES. 

COEFFICIENT 

MATERIAL. 

LIMIT. 

K. 

B. 

Elastic. 

Ultimate. 

ELASTICITY. 

White  Pine... 

Lbs. 

4,320 

Lbs. 
5,280 

Lbs. 
293 

0.86 

1.28 

Lbs. 

883,636 

Yellow  Pine.  . 
Locust.   .  .  . 

12,720 

8  4.OO 

16,740 
13  680 

930 
760 

0.84 
O.82 

1.96 

2  .  7O 

3,534,727 

2  046  315 

Black  Walnut. 

5.640 

7,44° 

413 

0.50 

0.72 

1,944,000 

White  Ash  ... 

6,360 

9,720 

540 

1.50 

2.50 

1,080,000 

White  Oak  ... 
Live  Oak  

7,200 

0.040 

9,840 
11,280 

547 
627 

0.90 

O.Q4 

1.76 

1.38 

1,620,000 
i,  8m,  428 

The  highest  values  are  as  often  given  by  green  as  by  sea- 
soned timber 

The  density  of  the  wood  does  not  determine  the  coeffi- 
cient ;  .  .  .  . 

A  high  coefficient  usually  accompanies  high  tenacity  and 
great  transverse  strength,  but  it  is  not  invariably  the  fact  that 
maximum  ultimate  strength  is  accompanied  by  initial  stiff- 
ness  

Ovaries  from  10,000  to  17,000  pounds  per  square  inch  (or 
B,  from  556  to  944)  with  a  mean  value  of  about  13,000  (or 
about  722  for  />). 

In  "Van  Nostrand's  Magazine"  for  Feb.,  1880,  Mr.  F.  E. 
Kidder,  B.C.E.,  gives  the  following  results  of  experiments  with 
5  yellow  pine  specimens  about  1.25  inches  square  in  section  and 
8  white  pine  specimens  about  1.5  inches  square;  all  on  sup- 
ports 40  inches  apart : 
34 


530 


FLEXURE  OF  SOLID  BEAMS. 


[Art.  63. 


Yellow  Pine. 

GREATEST.  MEAN. 

Coefficient  of  elasticity. . .   1,926,160  Ibs 1,821,630  Ibs. . . 

K. 14,654  Ibs 13, 048  Ibs... 

B 813  Ibs 725  Ibs. .. 


Coefficient  of  elasticity  . . 

K. 

B.. 


White  Pine. 

1,461,728  Ibs 1,388,497  Ibs... 

9,440  Ibs 8, 297  Ibs. .. 

524  Ibs. . . .  461  Ibs. . . 


LEAST. 

1,707,282  Ibs. 

12,280  Ibs. 

682  Ibs. 


1,251,252  Ibs. 

7, 573  Ibs. 

421  Ibs. 


Table  III.  contains  values  of  B  which  have  been  computed 
from  data  determined  by  MM.  Chevandier  and  Wertheim 
("  M£moire  sur  les  Propri£te"s  Me"caniques  du  Bois,"  1848).  The 
timber  was  from  the  Vosges.  The  great  variations  in  the 
length  of  span  and  dimensions  of  beam  render  these  especially 
valuable. 

TABLE  III. 

Vosges  Timber. 


BREADTH. 

DEPTH. 

SPAN. 

W  AT  CENTRE. 

B. 

Ins. 

Ins. 

Ft. 

Lbs. 

Lbs. 

r  11.4 

12.8 

42.64 

14,120 

339 

10.  0 

II.  2 

36.08 

11,867 

356 

.    8.8 

9.6 

29.52 

7,584 

287 

fal  6.7 

7-7 

29.52 

4,580 

355 

1*4 

3.65 

4-85 

29.52 

1,137 

415 

9-7 

2.16 

9.91 

2,017 

445 

,  9-5 

i.  ii 

9.91 

581 

500 

9.2 

10.9 

18.04 

17,356 

293 

8.6 

9-3 

18.04 

15,816 

392 

7.6 

8.6 

18.04 

11,495 

376 

6-3 

7-4 

18.04 

12,155 

643 

^ 

5-4 

6-3 

18.04 

4,895 

421 

o  " 

3.26 

3-2 

9.84 

1,188 

354 

3-07 

3-16 

8.20 

1,617 

433 

11.50 

2.15 

18.04 

957 

343 

5-64 

1.66 

9.84 

825 

532 

L  9-5 

i.  ii 

9.84 

715 

614 

Art.  63.] 


TIMBER. 


531 


The  weights  of  the  beams  were  allowed  for  in  the  manner 
already  shown  in  that  section  of  this  Art.  which  is  headed  "  Cop- 
per, Tin,  Zinc,  and  their  Alloys." 


TABLE   IV. 

Laslett  's    Tests. 
Sections  2x2  inches  with  span  of  6  feet. 


KIND    OF  TIMBER. 


W,  IN  LBS. 


B,  IN  LBS. 


FINAL  DEFLEC- 
TION. 


COEFFICIENT 
OF  ELAS.,  OR  E. 


Oak,  English 562 

Oak.  English 4°7 

Oak,  English 813 

Oak,  French 877 

Oak,  French  831 

Oak,  Tuscan  758 

Oak,  Sardinian 758 

Oak,  Dantzic 474 

Oak,  Spanish 562 

Oak,  American,  white 804 

Oak,  American,  Baltimore...  723 

Oak,  African  (or  teak) 1. 108 

Teak,  Moulmein 9*3 

Teak,  Moulmein  843 

Iron  wood,  Burmah 1,273 

Chow,  Borneo 975 

Greenheart,  Guiana — i,333 

Sabicu,  Cuba 1,293 

Mahogany,  Spanish 856 

Mahogany,  Honduras 802 

Mahogany,  Mexican 783 

Eucalyptus,  Australia  : 

Tewart 1,029 

mahogany 686 

iron-bark 1,407 

blue-gum 712 

Ash,  English 862 

Ash,  Canadian 638 

Elm.  English 393 

Rock  elm,  Canada 920 

Fir,  Dantzic 877 

Fir.  Riga 600 

Fir,  spruce,  Canada 670 

Larch,  Russia 626 

Cedar,  Cuba 560 

Red  pine.  Canada 653 

Yellow  pine,  Canada 627 

Yellow  pine,  Canada 483 

Yellow  pine,  Canada 304 

Pitch  pine,  American 1,049 

Pitch  pine,  American 930 

Pitch  pine,  American 744 

Kauri  pine,  New  Zealand 719 


422 

305 
oio 
658 
623 
569 


632 

955 

731 

1,000 

970 

642 

602 

587 

772 

515 

1.055 


479 
295 
690 
658 
450 
503 
470 
420 
490 
470 
362 
228 


558 
539 


Inches. 
5.10 

3-95 
7.71 
6.00 
7  58 
7.66 
6.50 
6.46 
6.62 
8.83 
7-13 


4  25 
2.83 
4.62 
3-75 
3-45 
4.06 
3-92 

4-75 
4.71 
3-8i 
4.21 
8.63 
7-37 

5  29 
8.79 
5-14 
3-63 
5-19 
4-33 
4-37 


3  39 
3-45 
4-79 
4.67 
4.42 
4.00 


Pounds. 


902,600 

1,536,800 

1,440,000 

605,000 

871,400 


1,184,600 
1,547,200 
1,010,880 
1,378,500 
1,172,400 
2,369.300 
2,472,300 
1,057,900 
2,369,300 
1,882,800 
1,187,100 
2,021,800 

1,791,000 

2.420.000 
1,805,100 
1,404,000 


1,299,700 
1,395.400 
1,763,200 
1,849,200 


2,030,800 
1,834,300 
1,602,000 
1,636,300 


E  has  been  computed  only  for  those  cases  in  which  W  exceeds  700. 


532  FLEXURE   OF  SOLID  BEAMS.  [Art.  63. 

In  the  cases  of  the  fir  specimens,  B  increases  very  con- 
siderably as  the  depth  of  the  beam  decreases,  and  with  little 
irregularity.  The  same  general  result  seems  to  hold  with  the 
oak  specimens,  although  there  are  very  marked  irregularities- 
On  the  whole,  therefore,  these  experiments  would  seem  to 
show  unmistakably  that  B  or  K  has  much  larger  values  for 
small  depths  of  beam  than  large. 

The  modulus  of  rupture,  K,  may  of  course  be  found  by  tak- 
ing I S£,  but  its  values  are  not  given  in  the  table. 

Tables  IV.,  V.,  VI..  and  VII.  contain  values  of  B  and  E 
which  have  been  computed  from  data  determined  by  the 
English  experiments  of  Messrs.  Laslett,  Maclure,  Fincham, 
Edwin  Clark  and  G.  Graham  Smith.  These  experiments  are 
among  the  latest  and  most  valuable  ever  made. 

In  all  these  tables  Wis  the  total  load  applied,  including  the 
weight  of  the  beam,  wherever  that  correction  is  made. 

In  Table  IV.  the  coefficient  of  elasticity  is  computed,  in  all 
cases,  for  a  centre  load  of  390  pounds.  In  Table  V.  the  centre 
load  for  the  same  computation  is  1,680  pounds  ;  and  in  Table 
VII.  the  elastic  load  had  different  values  for  different  beams. 

In  all  cases,  except  the  four  noted  in  Table  VII.,  the  ap- 
plied loads  were  placed  at  the  centre  of  the  span. 

Although  these  experiments  do  not  embrace  a  great  variety 
of  cross  section  for  all  kinds  of  timber,  yet  Tables  IV.,  VI.  and 
VII.  give  much  larger  values  of  B  for  small  depths  of  pine  and 
fir  beams  than  for  large  ones.  This  is  a  very  important  con- 
sideration in  connection  with  the  ultimate  resistance  of  beams, 
and  probably  obtains  for  all  kinds  of  timber.  In  fact,  Table 
III.,  as  has  been  observed,  indicates  the  same  results  for 
Vosges  fir  and  oak. 

These  experiments  also  showed  that  the  coefficient  of  elas- 
ticity, E,  varied  materially  in  the  same  specimen  for  different 
deflections,  and  that  values  among  the  greatest  may  be  found 
with  large  deflections  ;  also  that  the  "  elastic  limit "  for  flexure 
in  timber  beams  is  more  conventional  than  real,  since  with 


Art.  63.] 


TIMBER. 


533 


TABLE  V. 
Fin  chants  Tests. 

3x3  inches,  section  ;  4  feet  span  ;  very  dry  timber. 


KIND   OF   TIMBER. 

w. 

B. 

COEFFICIENT    OF  ELAS- 
TICITY. 

Riga  fir 

Pounds. 
4eao 

Pounds. 
67O 

Pounds. 
2  2Q'2,  76o 

Red  pine  

•1,780 

ceo 

I.^O*5,  OOO 

Yellow  pine    

2  71:6 

4.08 

i  550  ooo 

•7  2Q2 

487 

i  850  ooo 

Scotch  pine  

2  52O 

VT\ 

Q2<  OOO 

Kauri  pine     

J.  IIO 

608 

I  Q77  4OO 

TABLE  VI. 
Maclure's    Tests. 

Specimens  of  Memel  Fir. — 1849. 


FINAL   DEFLECTION, 

BREADTH. 

DEPTH. 

SPAN. 

W. 

B. 

INCHES. 

Inches. 

Inches. 

Feet. 

Pounds. 

Pounds. 

I 

I 

I* 

483 

644 

0-75 

I 

I 

Ii 

450 

600 

0.75 

2 

2 

«f 

I.QIO 

637 

I.OO 

2 

2 

2| 

1,311 

437 

1.125 

3 

3 

9 

1,104 

368 

3-5 

3 

3 

9 

1,482 

494 

4-5 

6 

12 

12 

34,720 

482 

2.0 

9 

12 

12 

38,080 

353 

2-5 

12 

12 

12 

6l,6oo 

428 

3-25 

534 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63, 


TABLE  VII. 
Tests  by  Edwin  Clark  and  G.   Graham  Smith. 


TIMBER. 

< 
• 
• 

i 

DEPTH. 

SPAN. 

W. 

B. 

FINAL   DE- 
FLECTION. 

COEFFICIENT 
OF  ELAS. 

Inches. 

Inches. 

Feet. 

Pounds. 

Lbs. 

Inches. 

Pounds. 

American  red  pine  

12 

12 

15 

33>497 

291 

4.00 

1,443,830  I 

\ 

American  red  pine  
American  red  pine  . 

12 

6 

12 
6 

15 
7-5 

29,908 
7070 
Distrib'd 

260 
256 

3.10 
1.68 

1,155,100 
1,015,900 

J 

u 

M 

Memel  fir  

13.5 

13.5 

10.5 

68,560 

293 



2,150,500 

Distrib'd 

I  -j.  e 

68  560 

_,  

Baltic  fir 

6 

12 

12.25 

19,145 

271 

.11 

1,573,400 

Baltic  fir  

6 

12 

12.25 

23,625 

335 

•93 

1,442,300 

Pitch  pine         

6 
6 

12    * 

12.25 

23,030 

326 
0-36 

.31 

3,125,000 

£ 

Pitch  pine 

Pitch  pine  .     . 

14 

ig 

448 

Pitch  pine 

1 

15 
12 

10.5 
12.25 

132,610 
16,800 

442 
238 

1,693,400 
1,247,000 

Red  pine  

Red  pine 

6 

j* 

J| 

Distrib'd 

L 

Quebec  yellow  pine  — 

H     . 

15 

IO-5   1 

68,600 

229 

— 

I.329.750 

G 

j 

Distrib'd 

d 

Quebec  yellow  pine  
Quebec  yellow  pine  
Quebec  yellow  pine  

M 

14 
H 

IS 
15 
15 

IO-5    I 
10.5 
10.5 

68,600 

85,792 
76,160 

229 
286 
254 

— 

1,329,750 
1,270,000 

loads  about  half  the  breaking  weight,  not  only  the  deflection 
but  the  "  set  "  varied  with  the  time. 

The  quantity  ordinarily  termed  the  load  at  the  "  elastic 
limit  "  may  be  taken  from  0.5  to  0.6  the  breaking  weight.  In 
Table  VII.  it  varied  from  0.50  to  0.78. 

The  latest  experiments  on  timber  beams  are  those  of  Col. 
Laidley  and  Prof.  Lanza;  both  experimented  during  1881. 
Col.  Laidley's  results  are  given  in  Table  Vila. 

As  was  to  be  expected,  in  accordance  with  conclusions 
already  drawn,  the  sticks  of  Oregon  pine  with  the  smallest 
depths  gave  values  of  K  and  B  considerably  larger  than  the 
others.  These  results  emphasize  the  fact  that  for  large  beams 
K  or  B  must  be  taken  from  tests  on  beams  equally  large  if 
accurate  computations  are  to  be  made.  With  these  consider- 


Art.  63.] 


TIMBER. 


535 


TABLE   Vila. 
Seasoned  Sticks,  Loaded  at  Centre. 


NO. 

KIND    OF  WOOD. 

z 

in 

WIDTH. 

DEPTH. 

K  -  i8£. 

LBS.  PER  SQ. 
INCH. 

B. 

REMARKS. 

Ins. 

Ins. 

Ins, 

->   4.8 

66  1 

• 

Oregon  pine  

22 

1  .22 

1.23 

13,210 

734 

Cross  grained. 

Oregon  pine 

16,570 

4 

Oregon  maple  

44 

3.63 

3-63 

10,560 

587 

California  laurel  

3  58 

8,920 

406 

Worm  eaten. 

6 

Ava  Mexicana  

44 

3.69 

3.69 

9.930 

552 

Oregon  ash  

3  64 

3.64 

8,460 

47° 

Cross  grained. 

8 

Mexican  white  mahogany. 

44 

3-77 

3-77 

9,6lO 

534 

Mexican  cedar 

Mexican  mahogany 

87Q 

TABLE 
Seasoned  Spruce  Beams. 


K  =  iZB. 

NO. 

SPAN. 

WIDTH. 

DEPTH. 

MANNER   OF  LOADING. 

LBS.  PER  SQUARE 

B. 

INCH. 

Feet. 

Inches. 

Inches. 

I 

15.00 

2.00 

12.00 

At  centre. 

5,526 

3°7 

2 

6.60 

2.00 

9.00 

*         ' 

5,389 

299 

3 

15.00 

2.00 

12.00 

'         ' 

5,237 

291 

4 

6.67 

2.75 

9.00 

*         • 

4,082 

226 

4.00 

3.00 

9.00 

*         fc 

3,285 

183 

6 

IO.OO 

3.00 

9.00 

'         ' 

4,508 

250 

7 

15.00 

3.00 

9.00 

'         ' 

5,651 

3M 

8 

20.00 

3-90 

12.00 

'         ' 

4,253 

237 

9 

10 

IO.OO 

16.00 

2.50 

3-75 

I3-50 
12.  OO 

4  .  5  feet  from  one  end. 

3,787 
3,271 

210 

182 

ii 

7.00 

7.00 

2.00 

At  centre. 

8,748 

486 

12 

7.00 

*-75 

6  75 

'         l 

7.562 

420 

T3 

6.67 

3.00 

9.00 

'         ' 

4,931 

2  74 

*4 
IS 

6.67 
16.00 

3-00 
3-90 

9  oo 

12.00 

At  4  points,  16  ins.  apart. 
4.5  feet  from  one  end. 

4,96l 

5,218 

276 

536 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


ations   in   view,  Prof.  Lanza's   results  for  large  spruce  beams, 
which  are  given  in  Table  VI \b.,  possess  great  value. 

With  the   exception  of  Nos.  u   and   12  the  material  was 
common  merchantable  lumber. 


Timber  Beams  of  Natural  and  Prepared  Wood. 

Table  Vllr.  contains  the  results  of  some  experiments  by 
A.  M.  Wellington,  C.E.  ("  R.  R.  Gazette,"  Dec.  17,  1880)  on 
small  specimens  ij^  inches  square  and  15  inches  between  sup- 
ports. "  All  the  woods,  except  as  specified,  had  been  cut  six 
to  eight  months  and  were  partially  seasoned." 


TABLE   Vile. 
Specimens  1.25  inches  square,  15  inches  long. 


KIND   OF  TIMBER. 

NATURAL. 

PREPARED. 

LOSS,  PER 
CENT. 

W,  in  Lbs. 

B,  in  Lbs. 

W,  in  Lbs. 

B,  in  Lbs. 

White  oak,  well  seasoned.  . 
White  ash  

989 
926 
864 
763 
941 

747 
742 
685 
628 

633 
593 
553 
489 
602 

479 
476 

439 
401 

~8^5 
801 
763 

755 

643 
640 
550 

527 
513 
489 
482 

411 
409 
332 

II.  2 

7-2 
0.0 
2O.  O 

Beech 

Elm     ' 

Pin  oak  

^White  oak    green 

Soft  maple 

13-7 
6.9 
17.2 

Black  ash   

Sycamore  

The  "  prepared  "  specimens  had  been  treated  by  the  Thil- 
meny  (sulphate  of  baryta)  process  ;  and  all  specimens  of  the 
same  kind  of  wood  were  cut  from  the  same  stick. 

The  column  "  Loss  "  is  the  per  cent,  of  loss  caused  by  the 
preservative  process  employed. 


Art.  63.]  CEMENT,   MORTAR  AND   CONCRETE.  537 


Cement,  Mortar  and  Concrete. 

Table  VIII.  and  Table  IX.  contain  values  of  K  computed 
from  data  given  by  Gen'l  Gillmore  in  his  "  Limes,  Hydraulic 
Cements  and  Mortars,"  1872.  All  the  prisms  were  2  inches 
square  in  cross  section  and  8  inches  long,  and  were  broken  by 
the  weight  W,  which  was  applied  at  the  centre  of  a  4-inch 
span.  K  is  computed  by  Eq.  (5),  all  dimensions  being  in 
inches.  The  composition  is  shown  in  the  tables.  The  pure 
mortars  of  Table  VIII.  were  kept  24  hours  in  a  damp  place, 
and  then  immersed  in  salt  water  until  broken.  Nos.  I,  2,  3 
and  4  were  59  days  old  ;  the  others,  320.  As  a  rule,  those 
which  set  under  pressure  were  considerably  stronger  than  the 
others. 

In  Table  IX.,  all  the  prisms  set  under  a  pressure  of  32 
pounds  per  square  inch,  and  were  kept  in  sea  water,  after  the 
first  24  hours,  until  broken. 

Many  reliable  experiments,  such  as  those  which  follow, 
show  that  when  masonry  is  built  in  a  strictly  first-class  manner, 
its  transverse  resistance  is  very  considerable. 

Table  X.  is  taken  from  a  paper  entitled  "  Notes  and  Ex- 
periments on  the  Use  and  Testing  of  Portland  Cement,"  by 
Wm.  W.  Maclay,  C.E.,  in  the  "  Trans.  Am.  Soc.  of  Civ.  Engrs.," 
1877. 

The  concrete  prisms  were  six  inches  square  in  cross  section 
and  two  feet  long,  and  rested  on  supports  one  foot  apart.  W 
was  applied  at  the  centre  of  the  span.  If  Wv  is  the  weight  of 
the  prism  whose  length  is  equal  to  the  span,  Eq.  (5)  becomes  : 


in  which  £,  //  and  /  are  to  be  taken  in  inches. 


538 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


S.' 

•s 

« 

p  *  j  i  i 

» 

« 

i 

H 

£ 

S 

fo. 

£ 

w 

H 

CO 

*' 

G  • 

!«  s  i  &  i 

i 

1 

i 

1 

| 

i 

12 

00 

!  i  1 

1  a  a 

8  8  8  8  8 

8 

8 

8 

8 

8 

8 

8 

8 

8  2  s 

S    8-    g 

CO 

. 

• 

j 

; 

POSITION  OF  MORTAR. 

i  S  S  jj  ! 

i  g  g  S  : 
?  f  ?  P  c 

1  •§  4  *  * 

>  >  >  o  vT 

VO  ^  ^  >  g 

i  «-•  IT  «-  ^ 

G  C  C  "O 
«  W  S  0  c 

s  a  a  a  rt 

and  water,  stiff  

and  water,  thin  

_G 
0) 

i 

5 

c 

fl 

•a 

and  water,  stiff  

and  water,  thin  

and  water,  very  thin. 

d 

1 

D 

1 

t) 

§ 

w  o  S  S  ^ 

*O  «O  "O  T3  § 

^2  ^  "}  J2  « 

o  o  o  o  Si 

;:t  1  i  1 

Pure  cement 

Pure  cement 

Pure  cement 

Pure  cement 

Pure  cement 

Pure  cemeni 

Pure  cemeni 

Pure  cement 

• 



• 

1 

i 

i 

i 

b 

1 

1 

Baxter. 

w 

M 

« 

G 

G 

2 

*a 

jfl 

o 

rt 
1 

s 

1 

1 

1 

1 

1 

B 

«-  w  «-.  v-  w 

^  ^  g|  ^  gj 

S  S  2  S  S 

rt  rt  n  rt  S 

ames  River 

*osendale,  H 

*rt 

g 

^cT 

IS 

p^ 

^osendale,  I- 

losendale,  E 

Q 

1 

R.osendale,  E 

i 

H 

H 

N 

H 

Art.  63.]  CEMENT,   MORTAR  AND   CONCRETE. 


539 


TABLE  IX. 

Section  of  Prisms  2  inches  square.      Supports  4  inches  apart. 


KIND  OF  CEMENT. 

Pure 
cement. 

i  vol.  cement, 
i  vol.  sand. 

T  vol.  cement. 
2  vols.  sand. 

English  Portland  (artificial)                                 "* 

716 

690 

410 

fa 

420 

Delafield  and  Baxter  (Rosendale)  

«  62, 

CIQ 

OQQ 

lt  Hoffman  "  Rosendale         .                       .... 

•3 

637 

456 

M 

rS-> 

Round  Top  Md                                        

>, 

4"5O 

Utica  111 

•o 

567 

Sheperdstown,  Va  

S 

560 

53 

338 

Akron   NY                                 . 

CO 

489 

Sandusky    Ohio 

54" 
416 

348 

IS 

Lawrenceville  Manf.  Co.  (Rosendale)  ^ 
Sandusky    Ohio 

3 

602 

683 

Kensington,  Conn  

•    & 

716 

•?8o 

Lawrence  Cement  Co.,  "  Hoffman  "  Brand.. 
Round  Top   Md 

0 

£ 

656 

684 

K,  POUNDS  PER  SQUARE  INCH. 


In  Mr.  Maclay's  experiments,  since  the  span  was  twelve 
inches  and  the  ends  overhung  six  inches,  K  was  computed  by 
the  formula : 


3   W[  _    W 
2  bl?  ^~~  12 


(17) 


Table  XL  contains  the  results  of  some  French  experiments 
cited  by  Gen.  Gillmore  in  his  "  Limes,  Hydraulic  Cements  and 
Mortars."  The  concrete  prisms  were  of  Boulogne  Portland 
cement,  about  5.91  inches  square  in  section,  and  broken  by  a 
weight  (W)  at  the  centre  of  a  span  of  about  31.5  inches.  K 
was  computed  by  Eq.  (16). 

Table  XII.  gives  the  results  of  trials  of  concrete  prisms  by 
Gen.  Totten,  in  June,  July  and  August,  1837,  the  prisms  hav- 


540 


FLEXURE   OF  SOLID  BEAMS. 


[Art.  63. 


TABLE   X. 
Concrete  Prisms  6"  x  6"  x  2'.     Supports  I  foot  apart. 


T. 

T. 

DISPOSITION    OF   PRISMS   AFTER 
BEING  MADE 

W. 

JT. 

Fahr. 

18° 
18° 

Fa.hr. 

4°I 

AO 

Placed  in  North  River.  ")   d 
Exposed  outside   .                & 

Pounds. 

525 
77t; 

Pounds. 

44 
60 

18° 

4O° 

Kept  indoors  

I  12^ 

Q4. 

18° 
18° 

98° 
08° 

Placed  in  North  River.       | 
Exposed  outside                (  •"* 

175 
<?2C 

15 
27 

18° 

q8° 

Kept  indoors                      f  ^ 

•7CO 

6q 

24° 

4O° 

Exposed  outside     .            ^ 

I  800 

ICQ 

24° 

07° 

800 

67 

q2° 

4O° 

-  ;:::::   s 

I  47^ 

12^ 

32° 

08° 

"              \< 

700 

*8 

'  '     J    *S 

All  prisms  were  of  Portland  cement  concrete  ;  I  vol.  cement,  2  vols.  sand,  5 
vols.  small  broken  stone. 

T  =  temperature  of  air  when  concrete  was  mixed. 
T'  =  temperature  of  concrete  when  mixed. 

ing  been  made  in  Dec.,  1836.  The  cement  was  from  Ulster 
Co.,  N.  Y.  The  lime  (slightly  hydraulic)  was  from  Fort 
Adams,  R.  I.,  where  the  tests  were  made.  W  (the  centre 
breaking  weight)  and  K  are  in  pounds. 

These  experimental  results  on  the  flexure  of  solid  beams  in 
cement,  cement  mortar  and  concrete,  in  connection  with  those 
of  Gen.  Gillmore  on  the  adhesion  of  bricks  and  cement  or  ce- 
ment mortar,  show  that  masonry  beams  may  have  considerable 
transverse  resistance ;  and  such  resistance  may  be  an  important 
element  of  strength  in  some  arches  or  similar  masonry  struct- 
ure. It  should  be  borne  in  mind,  however,  that  such  a  conclu- 
sion is  implicitly  based  on  the  assumption  of  perfect  manipu- 
lation of  the  cement  and  mortar  and  the  most  conscientious 
care  in  laying  the  masonry.  These  ends  were  attained  in  the 
test  specimens,  but  it  is  probably  safe  to  say  that  such  is  not 
the  case  even  in  what  is  termed  first-class  masonry. 


Art.  63.]  CEMENT,   MORTAR  AND   CONCRETE. 


541 


I  ^ 

3   ^ 

H       SJ 

a 

•te 

S 

I 


RESISTANCE  OF  CONCRETE. 

K,  in  pounds. 

1} 

co  O   CN   W   O  xn  O  O  r-^co  xn  M  rf  M  xn  r^«oo  co  O 
ex  O  xn  CN   OcoMOvr>T-t-cOMOOcor>.  Oco   o 

If 

O  co  u->o   i-i  coo   M  M   o  O  t^O  co  O  in  tr>  m  m 

CNCOtNMtNNMHI-l 

P 

M 

MOco  coo  O  cococococo  moo  N  t^«  r>.  »n  CM  M 

^ 

1 

Uittu|»na^t^ 

W,,  in  pounds. 

1     f 

O  ^  O  *™*  "^O  O  O  co  o  C3  f^»  ^  CN  o  O  M  r*^  o 
r^Ococo   OM  O   cocoo   ^tcoO   OCN  Oco  cor^ 
co  m  «^-  ^-  CN   O    O  ^O  O  in'st-^-cocococococo 

it 
"l 

co  CN  xno   O  ^co  co  co  oo   O  O  ^oo  CM  M  o  M 
O  CN  \O    •t'CO   xno    'fro   xn  ^t  co   ^  O   M   O    O  Oco 

O    CO  O  O  CO     OO    ^  *sf  CO  CO  CN    CN    CO  CN    CM    M    M    M 

M     M     M 

If 

r^OO   CNcoco   CN   rt-xnt-i   I^OCOM   ^••^•MVOO 
co   O   xnoi^t^oo    w   r^CN   TJ-O   Tj-w   M    O<"OI^ 

CONCRETES. 

•paonpojd  37943 

-UOD  JO  SUinjOA 

ft?     •^S$ZZ$ZS*ZSS?2 

1 

c 

-qsd  &M 

d 

CJ 

•KWK 

-,   ^~H^       ^H         «l   -h.             -i         c^«H»»       <*<> 

1 

•psonpojd  xe\ 
-aora  jo  aranioA 

O                   rj-              CN               xn                              O 

0                          CN                   M                   O                                          0 

«fl 

1 
c 

G 

.2 

1 

U 

-^ 

N                      CO             CO                xn               ^t               W 
\O                          Tj-                  CO                  CO                  CO                  CO 

6             6         o'          o'         6         o 

•PUBS  B3S 

M                               M                      M                      M                      M                      M 

,«*, 

-                     H.             *              H,             *              H. 

542 


FLEXURE  OF  SOLID  BEAMS. 


[Art.  63, 


•§• 


ON 


s      I 
X       i 

w    -5 
N 


oo-z 

OO'I 


OO'O 
OO'B 
OO'I 


Sz-o 
oS-i 

OO'I 


OO'O 
OO'I 


Sz'o 
oo'i 

OO'I 


OO'O 
OO'I 
OO'I 


£z-o       'atujq 
oS-o       'pucg 


Oi'O 

OO-I  'JU3UI33 


oo-o 

oo-o       'pucg 


O>  O  CO  M 
rf  N  O  O 
O  W  co  co 


O   -^-O   N 
co  co  m  rf 

M  M  co  a 


in  o^  r^  I  co  co  in 

rj-  M   rf  co   M   in 

O^  M  in  !->.  M  o_ 

ci        M"  '    M"        co 


^00     -l-O     ^  M     O  O   O 

O    t>i  r--.O  O    m  M   M    O^ 
WI-l  O  CrlWl^»P4v 


co        M         M         co 


oo-tt'-l-r^l  |ot->Ocom  r^co   rj-  o  co  N  r~  o  O 

r->.  r>-o   HI  w   r^  TJ-  r^>  ino  co    oco   rj-  M   M  O  co 

r^i-iOco  COWN       O"-iOwrtcoOwcoM 

cf       in  e^T        t-T       cT        co       in       cT       M" 


N  O  co  M  t^  co  r->i\o   o* 
•^O  co   >-•   M   co  rj-  <-(   rf> 

WMCNCO>-<MOCOO 

r?        rf        cf        in        M" 


r->.cor^c^cooo  inm 

Ococo    >-i   inrJ-M  O   rj- 

C4   i-i   co  r^  coco   M  co  M 

o"       in       M"  cf 


in       o         co        M 


llllljlfll 

c^PPMppNV^MCOc^PQ 


<N  CO 


Art.  63.]  STONE  BEAMS.  543 


Stone  Beams. 

But  few  experiments  have  been  made  on  the  transverse  re- 
sistance of  the  different  kinds  of  stone.  The  following  values 
of  K  have  been  computed  from  the  experiments  of  R.  G.  Hat- 
field  ("  Transverse  Strains  ")  and  Gen.  Gillmore  ("  Building 
Stones"). 

B.  K=  i85. 

Blue  Stone  Flagging 200  Ibs 3,600  Ibs. 

Sandstone 59  Ibs 1,062  Ibs. 


Brick,  common 33  Ibs 594  Ibs. 

Brick,  pressed 37  Ibs 666  Ibs. 

Marble,  Eastchester 147  Ibs 2,646  Ibs. 

Granite,  Millstone  Point  (doubtful) 133  Ibs 2,390  Ibs. 

Marble,  Eastchester 128  Ibs 2,300  Ibs. 

Granite,  Keene,  N.  H 103  Ibs 1,860  Ibs. 


Hatfield. 


Gillmore. 


All  beams  were  broken  by  centre  weights.  The  last  three 
tests  were  with  prisms  2  ins.  x  2  ins.  X  6  ins.,  over  a  span 
which  was  taken  at  3  inches. 

Practical  Formula  for  Solid  Beams. 

The  quantities  B,  K  and  E,  which  have  been  established, 
form  a  practical  basis  on  which  the  deflection  and  ultimate 
resistance  of  solid  beams  are  to  be  computed. 

Breaking  weight  (in  pounds]  at  centre  of  circular  beam,  Eq. 
(6): 


(.8) 
If  Wis  a  uniform  load  : 


In  Eqs.  (18)  and  (19),  A  (the  area),  r  (the  radius)  and  /  (the 
span)  are  to  be  taken  in  inches. 


544  FLEXURE  OF  SOLID  BEAMS.  [Art.  63. 

Breaking  weight  (in  pounds)  at  centre  of  rectangular  beams, 
£y.(S): 

...       •zKAh 


If  Wis  a  uniform  load  : 


In  Eqs.  (20)  and  (21),  A  (the  area),  b  (the  breadth),  d  (the 
depth)  and  /  (the  span),  are  to  be  taken  in  inches. 

If  /  is  expressed  in  feet,  and  all  other  dimensions  in  inches, 
Eq.  (20)  becomes  : 

W=     B^L=    Bb-^ (22) 

and  Eq.  (21): 

Deflection  (in  inches)  at  centre  of  circular  beams  : 

1£J    ^^Z    —      —     *"    "    - — — •          •          •          (2,A.} 

Deflection  (in  inches)  at  the  centre  of  rectangular  beams  : 

w  = 


In  Eqs.  (24)  and  (25),  Wis  the  centre  load,  and // the  total 
uniform  load,  expressed  in  pounds ;  while  A  (area),  /3  (cube  of 
span),  r  (radius),  b  (breadth),  and  d  (depth),  are  to  be  taken  in 
inches.  If  there  is  no  uniform  load,  *pl  is  zero ;  and  if  there  is 
no  centre  load,  Wis  zero. 


Art.  63.]  COMPARISON  OF  MODULI.  545 


Comparison  of  Modulus  of  Rupture  for  Bending  with  Ultimate 

Resistances. 

The  experiments  on  solid  beams  which  have  been  cited, 
show  the  somewhat  remarkable  result  that,  in  general,  K  has 
neither  the  value  of  the  ultimate  resistance  to -tension  nor  of 
that  to  compression  ;  nor,  indeed,  in  some  cases,  is  there  any- 
thing like  a  well  defined  relation  between  those  quantities.  If 
those  ultimate  resistances  have  widely  different  values,  K  may 
be  found  between  them  ;  in  other  cases  it  may  considerably 
exceed  either.  In  no  case,  however,  it  may  safely  be  asserted, 
will  it  be  found  less  than  both±  These  investigations  show  that 
K  varies  with  the  kind  of  cross  section,  and  it  is  altogether 
probable  that  it  also  varies  with  varying  proportions  of  the  same 
kind  of  cross  section.  Experimental  data  for  the  determina- 
tion of  this  point,  however,  are  still  lacking. 

In  the  absence  of  experiments  conducted  in  a  manner  proper 
to  the  solution  of  this  problem,  it  is  difficult  to  assign  confi- 
dently the  reason  for  the  facts  as  they  appear. 

The  explanation  will  probably  be  found  in  the  effects  of  the 
following  causes,  while  it  is  borne  in  mind  that  with  the  small 
ratios  of  span  to  depth  usually  found  in  connection  with  solid 
beams,  the  common  theory  of  flexure  is  only  loosely  approxi- 
mate, and  hence,  that  the  greatest  intensity  shown  by  the 
common  formulae  is  probably  considerably  different  from  the 
actual. 

The  varying  intensity  in  adjacent  fibres  prevents  perfect 
freedom  in  lateral  strains,  and  causes  a  corresponding  increase 
in  resistance.  In  the  experiments  which  have  been  made,  the 
place  of  greatest  intensity  of  stress  is  exceedingly  small,  thus 
placing  the  part  first  ruptured  somewhat  in  the  condition  of  a 
very  short  specimen.  Again,  after  the  elastic  limit  is  passed, 
in  consequence  of  the  flow  of  the  material,  it  is  highly  proba- 
ble that  the  law  of  the  variation  of  stress  intensity  changes  and 
35 


54-6  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

becomes  such  that,  with  the  same  greatest  intensity  at  the  sur- 
face of  the  solid  beam,  the  resisting  moment  is  considerably 
increased. 

Finally,  it  has  been  shown  that  the  experimentally  deter- 
mined ultimate  resistances  to  tension  and  compression  are, 
in  reality,  mean  intensities,  and  not  the  greatest  which  the 
material  is  capable  of  exerting  at  any  one  point,  or  along  any 
one  line,  as  in  the  extreme  fibres  of  a  bent  beam.  On  this 
ground  alone,  K  ought  to  be  considerably  greater  than  either 
T  or  C,  as  determined  from  the  usual  cross  sections. 


Art.  64. — Flanged  Beams  with  Unequal  Flanges. 

In  the  beams  which  are  to  follow,  the  material  is  distributed 
in  a  much  more  advantageous  manner,  in  respect  of  its  resist- 
ing moment,  than  in  the  solid  beams  which  have  been  hereto- 
fore treated.  In  these  bea-ms,  it  will  be  found,  in  almost  all 
cases,  that  the  ultimate  intensity  of  bending  stress,  at  the  point 
which  first  ruptures,  is  equal  either  to  the  ultimate  resistance 
to  tension  or  compression.  In  this  respect,  at  least,  therefore, 
the  ultimate  load  for  flanged  beams  is  more  easily  and  exactly 
determined  than  for  solid  ones. 

In  Fig.  i  is  shown  a  "flanged  beam."     The  "flanges"  are 

the   two    horizontal    parts    above   and 

i — j ^ 

— *-     below;  the  "web"  is  the  vertical  part 


uniting  the  two  flanges  so  as  to  form 
the  perfect  beam. 


In  order  that  there  may  be  economy 
.1  B  of  material  in  the  beam,  neither  flange 
)t      must  begin  to  fail  before  the  other  ;  in 
other  words,  the  two  exterior  layers  of 
F      fibres,  above  and  below,  must  begin  to 
~~FigT~  ,     fail  at  the  same  time.     . 

The  intensities,  then,  in  these  two 


Art.  64.]        EQUAL    COEFFICIENTS   OF  ELASTICITY.  S47 

layers  must,  at  the  instant  of  rupture,  equal  the  ultimate  re- 
sistances to  tension  and  compression  in  bending. 


Equal  Coefficients  of  Elasticity. 

By  the  common  theory  of  flexure,  if  the  two  coefficients  of 
elasticity  are  equal,  it  has  been  shown  that  if  C  is  the  centre  of 
gravity  of  the  cross  section,  the  neutral  axis  of  the  section  will 
pass  through  that  point.  Let  it  now  be  supposed  that  the 
lower  flange  is  in  tension  while  the  upper  is  in  compression. 
Also  let  T  represent  the  ultimate  resistance  to  tension  in  bend- 
ing, and  let  C  represent  the  same  quantity  for  compression  in 
bending.  Then,  since  intensities  vary  directly  as  distances 
from  the  neutral  axis, 

hi       T  T         ,,  ,  . 

'5         •'•         *  =  *       =  «*     ...     (i) 


The  ratio  between    ultimate   intensities    is  represented  by 
n'.     If  d  =  h  +  //x  is  the  total  depth  of  the  beam,  and  hence 


h  =  d  -  //, : 


.  =  *'(<*  ~  /'O  = 


If,  as  an  example,  for  mild  steel  there  be  taken  : 


The  relation  between  h  and  //,  shown  in  Eq.  (2)  is  entirely 
independent  of  the  form  of  cross  section.  But  according  to 
the  principles  just  given,  in  order  that  economy  of  material 


54-8  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

shall  obtain,  the  cross  section  should  be  so  designed  that  h  and  hi 
shall  represent  the  distances  of  the  centre  of  gravity  from  the  ex- 
terior fibres. 

The  analytical  expression  for  the  distance  of  the  centre  of 
gravity  from  DF  is  : 


2         (b  -  , 

b<d  +  (b  -  by  +  (b,  -  by, 

The  meaning  of  the  letters  used  is  fully  shown  in  the  figure. 
In  order  that  the  beam  shall  be  equally  strong  in  the  two 
flanges,  the  various  dimensions  of  the  beam  must  be  so  de- 
signed that 

*,?=*,    ........     (4) 

It  would  probably  be  found  far  more  convenient  to  cut  sec- 
tions out  of  stiff  manilla  paper  and  balance  them  upon  a  knife 
edge. 

The  moment  of  inertia  about  the  axis  AB,  thus  deter- 
mined, is  : 


This  value  is  to  be  substituted  in  Eq.  (2)  of  Art.  62,  now 
changed  to 

*=?=¥• 

h          /zx 

For  various  beams  whose  lengths  are  /  and  total  load  W, 
,    the  greatest  value  of  M  becomes  : 


Cantilever  uniformly  loaded  : 


Wl 
— 

2 


Art.  64.]        EQUAL   COEFFICIENTS  OF  ELASTICITY.  549 

Cantilever  loaded  at  end  : 

M  =  W  7. 
Beam  supported  at  each  end  and  uniformly  loaded: 


Beam  supported  at  each  end  and  loaded  at  centre  : 

•:.jf-.m, 

4 
The  last  two  cases  combined  : 


=liW  +  pl\ 

\        2         / 


Sometimes  the  resistance  of  the  web  is  omitted  from  con- 
sideration. In  such  a  case  the  intensity  of  stress  in  each  flange 
is  assumed  to  be  uniform  and  equal  to  either  T  or  C.  At  the 
same  time  the  lever  arms  of  the  different  fibres  are  taken  to  be 
uniform,  and  equal  to  h  for  one  flange  and  /^  for  the  other,  h 
and  7/j  now  representing  the  vertical  distances  from  the  neutral 
axis  to  the  centres  of  gravity  of  the  flanges,  while  d  =  h  -f  //,. 

On  these  assumptions,  if  f  is  the  area  of  the  upper  flange, 
and  f  that  of  the  lower,  there  will  result  : 

M  =  fC.h+fT.hl  ......    (5) 

But  since  the  case  is  one  of  pure  flexure  : 

fC  =  f'T  ............     (6) 

f'Td     ...    (7) 


550  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

Also,  from  Eq.  (6)  : 

f        T 

—  =  (&} 

f        C 


Or,  the  areas  of  the  flanges  are  inversely  as  the  ultimate  re- 
sistances. 

Unequal  Coefficients  of  Elasticity. 

All  these  results  presuppose  equality  between  the  coeffi- 
cients of  elasticity  for  tension  and  compression.  In  some  cases 
this  presumption  is  not  permissible.  To  the  formulae  of  Art. 
27  resort  must  then  be  made. 

The  neutral  surface  must  first  be  located.  If  d  is  the  total 
depth  of  the  beam,  /^  =  d  —  h ;  h,  then,  must  be  found.  Eq. 
(5)  of  Art.  27,  when  applied  to  Fig.  i;  becomes  : 


m, 


-  *-*.)• 


E'  representing  the  coefficient  of  elasticity  for  compression,  and 
E  that  quantity  for  tension. 

Performing  the  operations  indicated  and  reducing,  writing 
n  for  E'  -5-  E  : 


(n  -  \)b'h*  +  2\nt\b  -  b')  +  t,(b,  -  b')  +  b'<f]k 

=  nt\b  -  b'}  +  (2d  -  t,}  (b,  -  b'y,  +  b'd*  .       .     (9) 

h  is  to  be  measured  on  the  compression  side  of  the  beam. 
This  is  a  quadratic  equation  of  condition  for  the  determina- 
tion of  h.     It  is  best  to  leave  it  as  it  is  until  the  numerical  sub- 


Art.   64.]  UNEQUAL   COEFFICIENTS.  551 

stitutions  are  made  and  then  to  solve  it.     /*x  immediately  results 
from  the  equation  //x  =  d  —  h. 

Frequently  there  is  no  compression  flange,  the  section  being 
like  that  shown  in  Fig.  2.     In  such  a  case  b  is 
equal  to  b',  or  f  is  equal  to  zero  ;  hence  the  two 
terms  nt'(b  —  b')  and  nt'\b  —  b')  in  Eq.  (9)  disap- 
pear.    No  other  change  occurs.  ,  - 

Eq.  (i)  of  Art.  27  then  gives  the  following 
resisting  moment  of  the  section  : 


=-    t&  -(b 


C  is  the  greatest  intensity  of  stress  in  the  section  of  the 
same  kind  as  £'. 

If  the  section  is  like  Fig.  2,  b  again  equals  b'  and  the  term 
(b  —  b')  (Ji  —  /')3  in  Eq.  (10)  disappears,  but  nothing  else  is 
changed. 

If  T  is  the  greatest  stress  on  the  other  side  of  the  neutral 
surface  from  C: 

M  =        \_nbh*  -  n(b  -  b')  (h  -  tj  +  bji? 


In  order  that  the  beam  may  be  equally  strong  in  the  two 
flanges,  the  ratio  between  h  and  /*„  as  determined  by  Eq.  (9), 
should  be  the  same  as  that  determined  by  the  following  proc- 
ess. If  u  is  the  rate  of  strain  at  units'  distance  from  the  neu- 
tral surface  : 


552  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

Euk=C\         h        CE 

r     .V-T-   =    -^TCV        .        .         .         .         (12) 

*        TE 


If  there  is  no  waste  of  material,  the  cross  section  must  be 
so  designed  that  the  ratios  given  by  Eqs.  (9)  and  (12)  will  be 
the  same. 

If  the  thicknesses  of  the  flanges  /'  and  /,  are  small  com- 
pared with  the  depth  d  of  the  beam,  and  if  b'  also  is  small,  i.  e., 
if  the  flanges  are  assumed  to  give  the  whole  resistance  to  bend- 
ing while  the  web  takes  up  the  shear,  Eqs.  (10)  and  (11)  may 
be  much  simplified. 

C         T 
Making,  therefore,  b'  =  o  in  Eq.  (10),  putting  -  —  -  =  —  and 

then  expanding  the  quantities  (h  —  /')3  and  (h:  —  /\)3  : 


M  =  Cbf    A  -  /•  +          +  TV,    //,  -  /,  +  . 


t'2  t  2 

Under  the  conditions  taken,  Cbt'  =  Tbj^  ;  also,  —=•  and  —*- 
are  very  small  and  may  be  neglected.  Hence, 

M  =  Cbf  (d  -t'  -  t,}  =  TVi  (d  -  t'  -  /,)      .     (13) 

But  both  of  these  approximations  have  made  M  too  small. 
As  an  approximate  compensation,  therefore,  —  (  —  t-M  may 
be  written  for  —  (t'  +  /,).  The  moment  then  becomes  : 

M=  Cbt'    d-*-  .....     (14) 


The  quantity  within  the  parenthesis  of  the  second  member 
of  this  equation  is  evidently  the  distance  between  the  centres 


Art.  64.]  UNEQUAL  COEFFICIENTS.  553 

of  gravity  of  the  flanges,  while  the  quantity  Cbt'  =  Tbj»  is 
simply  the  flange  stress.  Eq.  (14)  is,  then,  identical  with  Eq. 
(7),  as  was  to  be  anticipated.  The  equality  of  flange  stresses 
gives  : 


a  relation  identical  with  Eq.  (8). 

If  desirable,  an  approximate  correction  for  the  neglect  of 
the  web  may  be  introduced  in  Eq.  (14).  It  has  been  seen  that 
that  equation  is  precisely  the  same  as  if  E'  were  equal  to  E> 
i.e.,  as  if  the  two  coefficients  of  elasticity  were  equal.  Now, 
it  will  be  shown  in  the  next  Article  that  if  E  '  =  £,  the  re- 
sistance of  the  web  to  bending  is  equal  to  that  of  one-sixth  of 
its  area  of  normal  section  concentrated  in  each  flange.  Hence, 
if  a  is  the  area  of  the  normal  section  of  the  web,  since  bt1  and  £,/, 
are  areas  of  the  normal  sections  of  the  upper  and  lower  flanges, 
there  may  be  approximately  written  : 


M  = 


Values  of  C  and  T  may  be  determined  by  experiment. 

In  the  case  of  solid  beams,  it  has  been  seen  that  if  r  and  r' 
are  certain  ratios,  K  =  rT  or  r  C.  Hence,  since  the  web  of  a 
flanged  beam  is  really  a  solid  beam  subjected  to  flexure,  Eq. 
(15)  may  be  written  : 


Jf=TD  (a'  +  ~)  =  CD  (a"  +  ™)   .    .    .    (16) 
In  which, 


554  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

t'  -\-  t 
D  =  d =  depth  between  flange  centres ; 

a'  =  bltl  —  area  of  bottom  flange  ; 
a"  —  bt'   =  area,  of  top  flange. 

X 

Cast-Iron  Flanged  Beams. 

In  the  preceding  Article  it  has  been  seen  that  r  is  equal  to 
about  2  for  a  solid  bar  with  square  cross  section.  This  would 
make  r  ~  6  =  %.  A  few  imperfect  experimental  indications, 
however,  seem  to  indicate  a  decrease  of  r  for  a  greater  ratio  of 
depth  to  breadth.  Let,  therefore,  r  -f-  6  =  0.25.  Eq.  (16) 
then  becomes : 


••TD(?'  +  ~) 07) 

If   W  =  centre  breaking  load  in  pounds  ; 

Wt  =  total  uniform  breaking  load  in  pounds  ; 

/      =  span  in  feet ; 
12  I      =  span  in  inches  : 

W  •   ill 

¥  V  \.  £(/  T  T  7*7 


TD 


In  the  same  manner 


(19) 


Or,  if //is  the  weight  of  the  beam,  supposed  uniformly  dis- 
tributed, 


Art.  64.]  CAST  IRON.  555 


TD 

__4       ....     (20) 

It  has  been  shown  under  the  head  of  "  Tension  "  that  T 
varies  from  15,000  pounds  per  square  inch,  for  ordinary  cast- 
ings, to  30,000  for  those  of  extra  quality.  In  Eqs.  (18),  (19) 
and  (20), 

D  must  be  taken  in  inches  ; 

a  and  a  in  square  inches  ;  and 

/  in  feet. 

Those  equations  have  been  verified  in  a  most  satisfactory 
manner  by  the  numerous  English  experiments  of  Hodgkinson 
and  Cubitt  ("  Experimental  Researches,"  etc.,  by  Eaton  Hodg- 
kinson, F.R.S.,  1846),  and  Berkley  ("  Proc.  Inst.  of  Civil 
Engineers,"  Vol.  XXX.),  as  is  shown  by  the  following  table. 
This  table  gives  the  actual  centre  breaking  weights  W,  of  the 
different  beams,  together  with  the  values  of  W  computed  by  the 
formula  of  Mr.  D.  K.  Clark  ("  Rules,  Tables  and  Data"),  which 
is  essentially  identical  with  Eq.  (18)  ;  Mr.  Clark  taking  the 
total  depth  minus  the  depth  of  the  lower  flange  instead  of 
"Z>,"  and  "  0.280,"  or  "  0.29^,"  instead  of  "  0.250." 

As  the  results  are  given  to  confirm  the  accuracy  of  the  for- 
mulae under  consideration,  they  are  stated  in  tons  of  2,240 
pounds.  Nos.  17,  27  and  34  were  of  the  form  shown  in  Fig. 
2  ;  the  others  had  sections  like  Fig.  I.  The  results  for  those 
three  beams  are  not  satisfactory,  and  Eq.  (10)  should  therefore 
be  used  in  all  such  cases  where  anything  more  than  a  very 
loose  approximation  is  desired.  In  that  Eq.  n  may  be  taken 
equal  to  unity,  on  account  of  the  great  irregularities  in  the 
ratio  of  the  two  coefficients  of  elasticity.  Since,  in  this  case 
(see  Fig.  i),  b  —  b'  Eq.  (10)  becomes  : 

M  =       16»  +  bji?  -  (b,  -  b'}  (It,  -  ty\  .    .    (21) 


556 


UNEQUAL-FLANGED  BEAMS. 


[Art.  64. 


Cast- Iron  Flanged  Beams. 


NO. 

SPAN. 

CENTRE  DEPTH. 

PROPORTION,   UPPER 
FLANGE   TO    LOWER. 

COMPUTED 

W  (TONS). 

ACTUAL 
JF(TONS). 

Feet. 

Inches. 

I 

4-5 

5-125 

to  I 

2.47 

2.98 

2 

4-5 

5.125 

tO  2 

3.27 

3-29 

3 

4-5 

5-125 

tO  4. 

3.83 

3.69 

4 

4-5 

5-125 

to  4 

3-37 

3.64 

5 

4-5 

5-125 

to  4.  5 

4.68 

4-79 

6 

4-5 

5-125 

to  4 

6.45 

6-46 

7 

4-5 

5.125 

to  5-5 

7-85 

7-47 

8 

4-5 

5-125 

to  3.2 

6.49 

6.7I 

9 

4-5 

5-125 

to  4-3 

8.04 

7-54 

10 

4-5 

5-125 

to  5.6 

9-56 

8.68 

ii 

4-5 

5-125 

to  6 

10.98 

'  11.65 

12 

4-5 

5-125 

to  7 

II.  OO 

10.40 

13 

4-5 

5-125 

to  6.7 

9.02 

9.40 

14 

7.0 

6.93 

to  6 

10.26 

9.90 

15 

7-o 

4.10 

to  6 

5-41 

6.05 

16 

9.0 

10.25 

to  8.3 

13.28 

12.80 

17 

4-5 

5-125 

none 

3.83 

3-93 

18 

4-5 

5-125 

to  4 

9.67 

10.00 

19 

4-5 

5-125 

to  4 

9.67 

IO.OO 

20 

4-5 

5-125 

to  5-5 

11.85 

11.75 

21 

4-5 

5-125 

to  5-5 

11.85 

11.85 

22 

4-5 

5-125 

to  7 

16.47 

14.25 

23 

4-5 

5-125 

to  7 

17.08 

18.00 

24 

18.0 

17.0 

to  4.6 

24-93 

25.00 

25 

11.67 

9.0 

to  1  .  33 

21.24 

20.00 

26 

27.4 

30.5 

tO  2.1 

94.64 

76.60 

27 

23.1 

36.1 

none 

330.00 

153-00 

28 

15-0 

7.15 

to  3.6 

7-75 

7.00 

29 

15-0 

7.17 

to  3.6 

7.96 

7.13 

30 

15-0 

10-75 

to  2.3 

1  1.  02 

11.50 

31 

15-0 

30-75 

to  2.3 

11.71 

12.00 

32 

15-0 

12.75 

to  2.7 

11-95 

10.25 

33 

15-0 

12.8 

to  2.25 

14.89 

15-75 

34 

15-0 

14.0 

none 

18.39 

12.38 

35 

15-0 

17.25 

to  2.2 

19-39 

16.00 

36 

7-5 

7-15 

103.4 

15-63 

15-63 

37 

7-5 

10.75 

to  2.25 

21.76 

23.87 

If  the  weight  of  the  beam  is  taken  into  consideration,  as  in 
Eq.  (20) : 


Art.  64.]  CAST  IRON.  557 


M=  (  W  +  f- 

2 


A  mean  of  three  of  Mr.  Hodgkinson's  beams  of  4.5  feet 
span,  5.125  inches  depth,  gave: 


~  =  8,766  Ibs.,        and       C  =  45,700  Ibs. 


One  of  Mr.  Cubitt's  beams  of  15  feet  span  and  14  inches 
depth,  gave  : 


—  =  28,100  Ibs.,       and       C  =  30,850  Ibs. 


The  bottom  flange  of  this  beam  was  unsound  : 
C  must  necessarily  depend  upon  the  span,  since  that  portion 
of  the  web  which  is  subjected  to  compression  is  somewhat  in 
the  condition  of  a  long  column.  This,  indeed,  is  true  of  the 
compression  flange  of  any  flanged  beam,  but  the  effects  result- 
ing from  such  a  condition  are  much  more  marked  in  the  class 
of  beams  shown  in  Fig.  2. 

If,  then,  W  is  the  centre  breaking  weight  and  Wl  the  total 
uniform  breaking  load  (not  including  the  weight  of  the  beam), 
Eq.  (21)  becomes: 


In  this  equation,  /  must  be  taken  in  feet  and  other  dimen- 
sions in  inches. 

For    5  feet  span  C  may  be  taken  at  45,000  Ibs. 
For  15  feet  span  C  may  be  taken  at  35,000  Ibs. 


UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

In  order  that  a  beam  with  top  and  bottom  flanges  may  give 
the  best  result,  i.e.,  reach  its  ultimate  resistance  in  each  flange 
at  the  same  time,  Mr.  Hodgkinson  found  that  the  area  of  the 
lower  flange  section  should  equal  about  six  times  that  of  the 
upper.  That  relation  has  been  anticipated  in  Eq.  (8). 


Deflection  of  Cast-Iron  Flanged  Beams. 

If  W  is  the  centre  load  in  pounds,  /  and  w  the  span  and 
centre  deflection,  respectively,  in  inches,  and  /  the  moment  of 
inertia  of  the  cross  section,  Eq.  (8)  of  Art.  24  gives  : 


E  = 


Or,  if  /  is  in  feet,  which  is  more  convenient  : 


A  mean  of  two  of  Mr.  Berkley's  beams  gave  : 

/  =  4.5  feet  ;  w  =  0.284  inch  ;    W  =  20,160  Ibs.  : 
/=  18.74.     Hence:  E  =  12,424,600  Ibs. 

A  mean  of  two  of  Mr.  Cubitt's  beams  gave  : 

7=15  feet  ;  w  =  0.465  inch  ;    W  =  1  1,200  Ibs.  ; 
/=  227.03.     Hence:  E  =  12,886,720  Ibs. 

The  four  preceding  beams  had  top  and  bottom  flanges,  as 
in  Fig.  i.  Another  of  Mr.  Cubitt's  beams,  without  top  flange, 
as  in  Fig  2,  gave  : 


Art.  64.]  WROUGHT  IRON.  559 

/  =  15  feet ;  w  =  0.41  inch  ;    W  =  13,440  Ibs. ; 
/  =  373.     Hence  :  E  —  10,679,400  Ibs. 

This  last  beam  had  a  defective  bottom  flange,  hence  there 
maybe  taken  without  essential  error: 

E  =  12,000,000  Ibs. 
Taking  I  in  feet,  Eq.  (24)  now  gives  for  the  centre  deflection : 


w  —  — 2—      — r (25) 

i, 000,0007 


in  which  W  is  either  the  centre  load,  or  five-eighths  (^ths)  the 
total  uniform  load,  as  the  case  may  be. 

The  formula  by  which  /is  to  be  computed  is  the  one  which 
immediately  follows  Eq.  (4). 


Wrought-Iron  T  Beams. 

The  wrought-iron  T  beam  is  the  most  important  beam  of 
that  material  with  unequal  flanges.  In  the  case  of  wrought 
iron  the  two  coefficients  of  elasticity  are  A 
essentially  equal  to  each  other ;  conse- 


i  i 

quently  the  axis  about  which  the  moment    ^37 — \       f  \^j 

of   inertia   of   the    section    is   to  be  taken  n       K 

passes  through  the  centre  of  gravity  of  the  L_^> 

latter. 


F 


Fig.3 

All   the  experiments  cited  in  this  sec- 
tion are  those  of  Sir  William  Fairbairn,  given  in  his  "  Useful 
Information  for  Engineers,"  first  series. 


5°*0  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

Experiment  /. 

A  section  of  the  beam  is  shown  in  Fig.  3.     It  was  composed 
of  two  2^-inch   LS  riveted  to  a  5    x    Y^- 
^   inch   plate.     AD  was  horizontal,  and   the 
flange,   BF,   downward ;    hence   F  was   in 
tension. 

W  —  centre  breaking  weight  =   3,409 
Ibs. 

/,  by  Eq.  (29)  of  Art.  49,  =  1.738. 
^  =  distance  of  .centre  of  gravity  from 
F  =  1.91  inches. 
Span  =  /  =  7  ft.  =  84  inches. 

K  =  T'  =  apparent  intensity  of  tensile  stress  at  F. 
Hence: 


K  =  r  =  =  78,400  Ibs. 

4-f 

Experiment  II. 

Beam  and  data  the  same  as  before,  except: 
W  =  7,750  Ibs. 

/  =  27  inches. 
Hence : 

K=  r  =  -^=  57,344  Ibs. 

Experiment  III. 

Beam  and  data  the  same  as  before,  except : 
BF  was  upward,  causing  compression  at  F. 

W  =  10,777  lbs- 
/  =  27  inches. 


VVV        OF   THE  '       \ 

DIVERSITY. 

/•      -,  ^x.  L,  A  te  __  -^•t  ^»»     / 

Art.  64.]  WROUGHT  IRON. 

K  =  C'  =  apparent  intensity  of  compressive  stress  at  F. 

Hence: 

K  =  C'  =  78,400  Ibs. 

Experiments  II.  and  III.  were  made  by  testing  portions  of 
the  same  beam  used  in  Experiment  I. 

Experiment  IV. 

A  section   of  the  beam    is  shown   in  Fig.  4.,  but  it  was 
tested  with  the  rib  or  web  upward,  as  shown  in  Fig.  2. 

AD  =  2.85  inches.         BF  =  2.5  inches. 
Thickness  of  rib  =  0.29  inch. 
Thickness  of  flange  =  0.375  inch. 
W7  =  3,019  Ibs.         /  =  48  inches. 
Xi  distance  of  centre  of  gravity  from  F  =  1.86  inches. 
/=  o.< 


Hence : 

K=  C  =--  ^-'  =  68, 100  Ibs. 

Experiment  V. 

Beam  and  data  same  as  for  IV.,  except 
Rib  was  downward,  as  shown  in  Fig.  4 : 

W=  3,153  Ibs. 
36 


562  UNEQUAL-FLANGED  BEAMS.  [Art.  64. 

Hence : 

K  =  T(  =  71,000  Ibs. 

In  all  Jhese  experiments  half  the  weight  of  the  beam  was 
included  in  W. 

These  results  show  that  the  apparent  ultimate  intensities  of 
resistance  to  compression  and  tension  in  bending  of  T  beams 
may  be  taken  equal  to  each  other;  also  that  there  may  be 
taken : 

K  =  C  =  T  =  70,000  Ibs.  per  sq.  in. 

The  ultimate  tensile  resistance  (T)  of  this  iron  probably 
ranged  from  45,000  to  50,000  pounds  per  square  inch.  Hence, 
nearly : 


From  the  equality  of  C'  and  7"',  it  follows  that  the  beam  is 
equally  strong  whether  the  web  or  rib  is  up  or  down. 

Deflection  of  Wronght-Iron  T  Beams. 

If  w  is  the  centre  deflection  of  a  beam  loaded  with  the 
centre  weight  W,  E  the  coefficient  of  transverse  elasticity,  and 
/  the  span,  then,  as  has  been  seen  : 


or, 


'-•& <* 

A  mean  of  the  experiments  II.  and  III.  gave :    . 


Art.  64.]  WROUGHT  IRON.  563 

O.I  7  +  O  1  8 

W  —  4,040  Ibs.,         w  =  -     —  !  —     -  =  0.175  inches. 

/=  L738. 
Hence: 


This  is  a  small  value  for  E,  but  is  due  to  the  fact  that  the 
beam  was  a  built  one.* 

A  mean  of  the  experiments  IV.  and  V.  give: 

W=  1,400  Ibs.,      w  =  °'135  "*  -^-^  =  0.15025  in.,      /=  0.989. 

Hence: 

E  =  21,706,000. 

This  last  value  of  E  is  about  four  times  as  large  as  the 
other.  Hence  the  rolled  beam  would  deflect  only  one-quarter 
as  much  as  the  built  one.  All  values  of  W  were  within  the 
elastic  limit. 

These  values  of  E,  inserted  in  Eq.  (26),  will  give  the  deflec- 
tion for  a  load  W  (including  five-eighths  the  weight  of  the 
beam)  at  the  centre.  If  W^  is  the  total  uniform  load,  ^  Wl  is 
to  be  put  for  W  in  the  equation.  Eq.  (26)  requires  /,  w  and  / 
to  be  in  inches. 

If,  however,  /is  in  feet  and  other  dimensions  in  inches: 

36^/3 

W  =  -ET  .......    (28> 

The  foregoing  formulae,  both  for  breaking  weight  and  deflec- 

*  It  is  probable  that  the  riveting  was  done  by  hand.  The  improved  modern 
machine  riveting  would  make  a  much  stiffer  beam. 


564 


EQUAL-FLANGED  SEAMS. 


[Art.  65. 


tion,  may  be  used  for  the  bending  of  angle  irons  with  sufficient 
accuracy  for  all  ordinary  purposes. 


Art.  65. — Flanged  Beams  with  Equal  Flanges. 

Nearly  all  the  flanged  beams  used  in  engineering  practice 
are  composed  of  a  web  and  two  equal  flanges.  It  has  already 
been  seen  that  the  ultimate  resistances,  T  and  C,  of  wrought 
iron,  to  tension  and  compression  are  essentially  equal  to  each 
other;  the  same  may  be  said  also  of  its  coefficients  of  elastic- 
ity. While  these  observations  may  not  be  applied  with  pre- 
cisely equal  force  or  truth  to  the  milder  forms  of  steel  now 
working  their  way,  to  a  considerable  extent,  into  engineering 
construction,  they  certainly  hold  without  essential  error. 

In  Fig.  i  is  represented  the  normal  cross  section  of  an  equal- 
flanged  beam.  It  also  represents  what  may 
be  taken  as  the  section  of  any  wrought 
iron  or  steel  I  beam.  Although  the  thick- 
ness /'  of  the  flanges  of  such  beams  is  not 
uniform,  such  a  mean  value  may  be  taken 
as  will  cause  the  transformed  section  of 
-  Fig.  i  to  be  of  the  same  area  as  the  orig- 
inal section. 

Unless  in  very  exceptional  cases  where 
local  circumstances  compel  otherwise,  the 
beam  is  always  placed  with  the  web  ver- 
tical, since  the  resistance  to  bending  is 
much  greater  in  that  position.  The  neu- 
tral axis  HBw\\\  then  be  at  half  the  depth  of  the  beam.  Tak- 
ing the  dimensions  as  shown  in  Fig.  i,  the  moment  of  inertia 
of  the  cross  section  about  the  axis  HB,  is : 


12 


[Art.  65.  FORMULAE.  565 

while  the  moment  of  inertia  about  CD  has  the  value  : 


=  ....... 

12 

With  these  values  of  the  moment  of   inertia,  the  general 
formula,  M  =  —  ,  becomes  (remembering  that  </x  =  -  or  -)  : 


M=C  ......    (3) 

Or; 


C  is  written  for  K,  since  K  —  T  =  C. 

Eq.  (3)  is  the  only  formula  of  much  real  value.  It  will  be 
found  very  useful  in  making  comparisons  with  the  results  of  a 
simpler  formula  to  be  immediately  developed. 

Let  d^  =  y2  (d  -f-  //).     Since  t'  is  small,  compared  with  —  , 

the  intensity  of  stress  may  be  considered  constant  in  each 
flange  without  much  error.  In  such  a  case  the  total  stress  in 
each  flange  will  be  :  Cbt'  =  Tbt',  and  each  of  those  forces  will 
act  with  the  lever  arm  ^^  .  Hence  the  moment  of  resistance 
of  both  flanges  will  be  : 

Cbt'  .d,. 

The  moment  of  inertia  of  the  web  will  be  :   -  -  .     Conse- 

12 

quently,  its  moment  of  resistance  will  have  very  nearly  the 
value  : 

Ct/f 

*      —     6    '  . 


$66  EQUAL-FLANGED  BEAMS.  [Art.  65. 

The  resisting  moment  of  the  whole  beam  will  then  be  : 

......    (5) 


A  still  further  approximation  is  frequently  made  by  writing 
dji  for  k2  ;  then  if  each  flange  area  bt'  —  /,  Eq.  (5)  takes  the 
form: 


(6) 


Eq.  (6)  shows  that  the  resistance  of  the  web  is  equivalent  to 
that  of  one-sixth  the  same  amount  concentrated  in  each  flange., 

If  the  web  is  very  thin,  so  that  its  resistance  may  be  neg- 
lected : 


Or: 

M 


Cases  in  which  these  formulae  are  admissible  will  be  given 
hereafter.  It  virtually  involves  the  assumption  that  the  web 
is  used  wholly  in  resisting  the  shear,  while  the  flanges  resist  the 
whole  bending  and  nothing  else.  In  other  words,  the  web  is 
assumed  to  take  the  place  of  the  neutral  surface  in  the  solid 
beam,  while  the  direct  resistance  to  tension  and  compression 
of  the  longitudinal  fibres  of  the  latter  is  entirely  supplied  by 
the  flanges. 

Again  recapitulating  the  'greatest  moments  in  the  more 
commonly  occurring  cases  : 

Cantilever  uniformly  loaded  : 


Art.  65.]  FORMULA.  567 

Cantilever  loaded  at  the  end  : 
M  =  WL 

Beam  supported  at  each  end  and  uniformly  loaded : 

Wl       pi* 

^  =  -8~  =  T- 

Beam  supported  at  each  end  and  loaded  at  centre  : 

Wl 
M  = . 


Beam  supported  at  each  end  and  loaded  both  uniformly  and 
at  centre  : 


In  all  cases  W  is  the  total  load  or  single  load,  while  /,  as 
usual,  is  the  intensity  of  uniform  load,  and  /  the  length  of  the 
beam. 

In  "  Useful  Information  for  Architects,  Engineers  and 
Workers  in  Wrought  Iron,"  issued  by  the  Phoenix  Iron  Co.  of 
Phcenixville,  Penn.,  are  the  record  of  some  experiments  by 
which  the  value  of  C  or  T  may  be  determined.  These  will  now 
be  used. 

Example  I. 

A  /-inch  I  was  subjected  to  successive  loads  at  the  centre 
of  the  span,  the  ends  being  simply  supported.  The  beam 
weighed  60  pounds  per  yard;  consequently  the  area  of  the 
cross  section  was  6  square  inches.  The  span  was  21  feet,  or 
252  inches.  The  dimensions  represented  in  Fig.  i  are  the  fol- 
lowing : 


568 


EYE  BEAMS. 


[Art.  65, 


t   —  0.36  inches. 

h  —  5.95       "  .'.     fe  —  210.63. 

d  =  7.00       "  /.     d*  =  343. 

b  =  3-67       " 
(b  -  t)  =  3.31       " 
/'  =  0.525     " 

d^  =  y2  (d-\-  h}  —  6.475  inches. 
/=*•  -1.927      " 


The  following  table  gives  all  the  recorded  results. 


CENTRE 
LOAD. 

Lbs. 

DEFLEC- 
TION. 

PERMA- 
NENT SET. 

REMARKS. 

/3 

'  +  -§-/')  . 

W~*EI    l* 

Ins. 

Ins. 

Ins. 

2,000 

0.468 

w  =  o. 

537 

3,000 

0-743 

w  =  o. 

775 

4,000 
5,ooo 
6,000 
7,000 
8,000 

1.020 

1.298 

1.578 
1.887 
2.300 

0.029 
0.030 
O.O6O 
0.183 

Weight  removed.  .  . 
n             it 

W  =   I.OI2 

10  =  1.250 
The  coefficient  of  elasticity, 
£,   is  taken  at  30,000,000 
Ibs. 

9,000 

3-540 

5.298 

10,000 

With  the  load  of  10,000  pounds  at  the  centre  the  "  beam 
sunk  slowly,  top  flange  yielding."  The  beam,  therefore,  may 
be  considered  as  essentially  failing  with  a  load  of  10,000  pounds 
at  its  middle  point.  As  the  top  flange  yielded,  the  ultimate 
resistance  to  compression,  or  C,  will  be  given  by  the  experi- 
ment. 

In  reality,  the  beam  carried  a  uniform  load  of  20  pounds 
per  foot  (its  own  weight),  besides  the  single  load  of  10,000 
pounds  at  the  centre.  Hence,  Eq.  (22)  of  Art.  24  will  give  the 
value  of  M.  It  is  as  follows  : 


Art.  65.]  EXPERIMENTS.  569 


But  —  =  20  X  21  -•-  2  =  210;   PF  —   io,ooo,  and  /  =  252. 

/  is  taken  in  inches  because  the  dimensions  of  the  cross  section 
are  in  the  same  unit.     These  values  give  : 

M  =  643,230. 
Also  the  data  given  above,  placed  in  Eq.  (3),  give  : 

M=  C  x   13.37. 
Equating  these  values  : 

C  ••-  643,230  ~  13.37  =  48,110.00  pounds      .     .     (8) 

Again,  the  proper  data  inserted  in  Eq.  (6),  the  approximate 
formula,  give  : 

M  =  C  x  14.79. 

Hence  : 

• 
C  =  643,230  -5-  14.79  =  43490-00  pounds     .     .     (9) 

The  first  permanent  set  was  observed  with  a  centre  load  of 
5,000.00  pounds.     This  gives  a  bending  moment  at  centre  of 


M=         5>ooo  +  y    =  328,230. 

Hence  : 

C  =  328,230  -r-  13.37  =  24,550  pounds. 

As  the  permanent  set  with  this  load  was  very  small,  and  as 


570  EYE  BEAMS.  [Art.  65. 

there  was  none  at  all  observed  with  a  centre  load  of  4,000 
pounds  (nearly  corresponding  to  C  =  20,000.00  pounds),  the 
limit  of  elasticity  may  be  taken  at  about  : 

20,000   4-   24,000 

-  =  22,000.00  pounds. 


In  the  right  hand  column  of  the  table  are  calculated  the 
deflections  by  Eq.  (21)  of  Art.  24,  the  coefficient  of  elasticity 
being  taken  at  30,000,000.00  pounds.  By  Eq.  (i),  using  the 
data  already  given  : 

/  =  46.795. 
Hence  : 

/3  ~  48jE7  =  0.0002375. 
Also: 

|// -262.5.' 

These  values  inserted  in  the  formula  give  the  results  shown  in 
the  table.  The  experimental  quantities  are  seen  to  increase 
much  more  rapidly  than  the  results  given  by  the  formula.  The 

agreement,  however,  is  sufficiently  close  for  ordinary  purposes. 

• 

Example  II. 

The  second  example,  derived  from  the  same  source  as  the 
first,  is  that  of  a  9-inch  I,  87  pounds  per  yard.  The  data  to  be 
used  in  connection  with  Fig.  I  are  as  follows  : 

f  =  0.72  inches. 

b  =  4.00       "  .-.     /  —  It'  —  2.88. 

/  =  0.39       « 

d  —  9.00       "  /.     d*  —  729.000. 

//  =  7.56       "  .-.     /*3  —  432.581. 

(£  —  /)  =  3.61 


Art.  65.] 


EXPERIMENTS. 


571 


/    =  21  feet  =252  inches  ;    p  —  29  pounds  per  foot. 
d^  =  8.28  inches.  W  =  17,500.00  pounds. 

The  bending  moment  at  centre,  as  before,  is  : 

+          =  I'I2I'683-5- 


The  above  data  inserted  in  Eq.  (3)  give  : 

M  —  C  x  25.08. 
Hence  : 

C  —  1,121,683.5  -T-  25.08  =  44,724.00  pounds. 
Again  the  approximate  formula  Eq.  (6)  gives  : 

M  =  C  x  27.92. 
Hence  : 

C  =  1,121,683.5  -v-  27.92  =  40,175.00  pounds. 


(10) 


(11) 


The  results  of  this   experiment  are  given  in  the  following 
table,  exactly  as  in  Ex.  I. 


CENTRE 
LOAD. 

DEFLEC- 
TION. 

PERMA- 
NENT SET. 

REMARKS. 

~&  (-+H- 

Lbs. 

Ins. 

Ins. 

Ins. 

2,000 

0.228 

0.257 

4,OOO 

0.474 

0-454 

6,000 

o.  720 

0.651 

8,000 

O.g62 

0.848 

10,000 

1  .2OT 

0.048 

Weight  removed. 

1.045 

12,000 

1.432 

0.050 

"             *• 

E  is  taken  at  30,000,000.00 

13,000 

1.530 

O.II7 

•  »             ii 

Ibs. 

14,000 

1.863 

0.269 

<  t             ii 

16,000 

3-256 

I7,OOO 

5-233 

17,500 

5.602 

572  EYE  BEAMS.  [Art.  65. 

The  beam  may  be  considered  as  having  yielded,  in  failure, 
with  a  centre  load  of  17,500.00  pounds.  That  number  was 
consequently  taken  above  in  the  greatest  value  of  M. 

If  it  be  assumed  that  permanent  set  was  just  at  the  point 
of  beginning  with  the  centre  load  of  9,000.00  pounds,  which 
cannot  be  far  wrong,  the  corresponding  moment  will  be  : 


=L  (9, 


,000  +  =  586,152 

.-.     C  =  586,152  ~  25.08  =  23,370.00  pounds  (limit  of  elas.). 

Taking  a  mean  of  the  results  of  the  two  examples  : 
By  exact  formula  [Eq.  (3)]  : 

C  =  46,417.00  pounds. 

By  app.  formula  [Eq.  (6)]  : 

C,  =  41,833.00  pounds. 

For  the  limit  of  elasticity  : 

Ce  —  22,700.00  pounds  (nearly). 

These  results  may  be  considered  accurate  for  the  Phcenix 
Iron  Co.'s  beams.  These  experiments  were  made  in  1858. 

It  is  interesting  to  notice  that  these  beams  failed  in  the 
compression  flanges. 

It  is  also  important  to  observe  that  the  ultimate  resistance, 
C,  is  fully  equal  to  the  ultimate  tensile  resistance  of  good 
wrought  iron  in  large  bars.  This  serves  to  confirm  the  opin- 
ion that  the  ultimate  tensile  and  compressive  resistances  of 
wrought  iron  are  not  far,  at  most,  from  being  equal  to  each 
other,  and  that  these  quantities  may  be  used  for  C  or  K  in  the 


Art.  65].  EXPERIMENTS.  573. 

formulae  for  flanged  beams.  If  the  approximate  formula,  Eq. 
(6),  is  used,  however,  according  to  these  results  C  or  K  should 
be  taken  about  0.90  (nine-tenths)  of  the  value  used  in  the 
exact  formula,  Eq.  (3). 

The  last  column  of  the  second  table  is  calculated  by  the 
formula,  as  shown,  taking  E  at  30,000,000.00  pounds.  The 
same  general  observations  apply  to  these  results  as  in  the 
preceding  example. 


Example  III. 

The  data  for  this  example  are  taken  from  the  hand-book 
for  1 88 1  published  by  the  N.  J.  Steel  and  Iron  Co.,  Trenton, 
N.  J.,  where  the  beams  were  broken.  The  breaking  weight  is 
the  mean  of  two  results  for  light  6-inch  wrought  iron  Is. 

d  —  6.00  ins.  t  =  0.25  in.  /'  =  0.456  in. 

/  =  12  ft.  =  144  ins.  7  =  23.815,  by  Eq.  (i). 

Since  the  beam  weighed  40  pounds  per  yard  : 
W  =  14,000  -f  80  —  14,080  Ibs.  (centre  breaking  load). 

Hence  : 

~       Md 

C  -    — j  ==  63,840  Ibs.  per  square  inch. 

By  approximate  formula  : 

J   =  0.21.  /=   1.368      A      ^  +/=   1.573. 

di  =  5.544  ins.  M  =  506,880. 


574  EYE  BEAMS.  [Alt.  65. 

Hence,  by  Eq.  (6)  : 

Cl  —  57>93°  Iks.  Per  square  inch. 

Example  IV. 

A  9-inch  heavy  Trenton  beam,  85  pounds  per  yard.  The 
data  are  taken  from  the  same  source  as  were  those  in  Ex. 
III. 

d  =  9.00  ins.  t  =.0.38  in.  t'  =  0.68  in. 

/=  15  ft.  —  180  ins.         /—  108.47,  by  Eq.  (i). 

W  =  32,0x30  -j-  212  =  32,212  Ibs.  (at  centre). 

Hence  : 

~       Md        .. 

C  —  — Y  =  60,120  Ibs.  per  square  inch. 

By  approximate  formula  : 

~  =  0.484.  /=  2.72      /.     ~  +/=  3.204. 

• 

</x  =  8.32  ins.  M  =  1,449,540. 

Hence  by  Eq.  (6)  : 

^x  =  54>37O  Ibs.  per  square  inch. 
Taking  the  means  of  these  two  sets  of  results  : 


Art.  65.]  U.    S.    TEST  BOARD'S  EXPERIMENTS.  575 

.      By  exact  formula  [Eq.  (3)]  : 

C  =  61,980. 

By  app.  formula  [Eq.  (6)]  : 

C,  =  56,  ISO- 
All  the  conclusions  reached  in  connection  with  Exs.  I.  and 
II.  are  confirmed  by  the  results  of  Exs.  III.  and  IV. 

C  and  Ct  are  much  larger,  however,  for  the  Trenton  than  for 
the  Phcenix  beams,  and  both  are  very  high  for  beams  of  such 
length  of  span  with  no  lateral  support  for  the  compression 
flange. 

In  calculating  the  deflection  of  rolled  wrought-iron  beams 
E  may  be  taken  from  28,000,000  to  30,000,000. 

The  exact  formulae  of  this  Article  are  strictly  applicable  to 
rolled  beams  only,  but  the  approximate  formula  finds  exten- 
sive application  in  cases  of  built  beams. 

Experiments  by  U.  S.  Test  Board. 

Table  I.  contains  the  results  of  a  valuable  series  of  tests  by 
the  U.  S.  Board,  "  Ex.  Doc.  23,  House  of  Rep.,  46th  Congress, 
2d  Session." 

The  values  of  K  and  E  at  elastic  limit  are  computed  from 
data  contained  in  that  document  in  the  manner  already  shown 
in  detail,  and  which  it  is  not  necessary  to  repeat.  It  is  both 
interesting  and  important  to  observe  the  considerable,  though 
irregular,  increase  of  the  intensity  of  stress  in  the  exterior  fibre, 
at  the  elastic  limit,  with  the  decrease  of  depth.  E  is  seen  to 
vary  from  26,099,400  to  36,664,400,  with  a  mean  value  of 
31,128,260.  As  a  general  result,  E  is  slightly  larger  for  the 
smaller  beams  than  for  the  larger. 


5/6 


EYE  BEAMS. 


[Art.  65. 


•£H3d    NI    NVJS 

In  this  table,  /  is  exact.  Hence  K  is  computed  by  the  exact  formula  :  K  =  ~  .  All  beams  were  loaded  at  centre,  and  deflections 
were  taken  at  same  point.  _ 

CNCSNNNNHMNNHH«C,NHHNHHHMMMH 

•HDNI  aavnos 

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K^  IN  POUNDS  PER  SQUARE  INCH  AT 

Final  Load  and  Deflection. 

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fll'jf^^.MMMft*?*^!!.^? 

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jo    vaav    aaifidNoa 

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»0  O  "&OO   O\  ON  ONOO  00  CO  00  IN  t^  tx  tx  txVO  VO  VO  >O  VO  »O  •*  '*• 

Art.  65.] 


TESTS  OF  COL.    LAIDLEY. 


577 


A  chemical  analysis  of  six  specimens  from  these  beams 
gave  the  following  results. 

These  experiments  were  conducted  by  Gen'l  Wm.  Sooy 
Smith,  who  kindly  gave  to  the  writer  the  final  centre  loads  and 
deflections. 


PERCENTAGES     OF 


Sulphur. 

Phosph'us. 

Silicon. 

Total  Carb. 

Manganese. 

Copper. 

Cobalt. 

Nickel. 

O.OJO 

0.436 

0.189 

0.031 

0.031 

0.012 

0.029 

0.029 

0.008 

0.447 

0.190 

0.038 

0.028 

0.008 

0.021 

0.023 

O.OIO 

°-453 

.    0.203 

0.037 

0.028 

O.OIO 

0.015 

0.018 

O.OI2 

0.423 

0.182 

0.039 

0.022 

O.O22 

O.OIO 

0.015 

O.OO5 

0.271 

0.177 

0.027 

O.O28 

0.052 

0.031 

0.026 

O.OII 

°-375 

0.197 

0.039 

0.028 

O.OIO 

0.018 

0.016 

Col.  T.  T.  S.  Laidley,  U.  S.  A.,  has  also  completed  the 
tests  of  a  few  beams  to  failure,  the  results  of  which  are  given 
in  "Ex.  Doc.  12,  4/th  Congress,  1st  Session."  Table  II.  con- 
tains values  of  K  at  failure,  computed  from  Col.  Laidley's 
data.  Beams  No.  I  carried  a  load  uniform  from  end  to  end  by 

TABLE    II. 


DEPTH, 

SPAN    IN 

LATERAL  SUP- 

TOTAL LOAD 

A"  IN  LBS. 

NO. 

MEAN    OF 

LOAD. 

INS. 

FEET. 

PORT. 

IN  POUNDS. 

PER  SQ.  IN. 

I 

15-0 

28.5 

3  Exps. 

Uniform. 

Uniform. 

Il8,OOO 

54,260 

2 

15-0 

28.0 

2  Exps. 

Centre. 

None. 

28,650 

25,890 

3 

10.5 

17.0 

2  Exps. 

Centre. 

None. 

2I.O2O 

32,2IO 

4 

10.5 

17.0 

I  Exp. 

Centre. 

None. 

22,O2O 

38,O7O 

37 


5  78  B  UIL  T  BEAMS.  [Art.  66. 

means  of  brick  masonry  arches,  which  thus  also  gave  to  them 
a  uniform  lateral  support.  This  lateral  support  produced  a 
very  high  value  of  K,  i.e.,  54,260  pounds,  which  fell  to  25,890 
with  no  lateral  support.  In  the  latter  case  nothing  prevented 
the  compression  flange  yielding  laterally  like  a  column.  The 
lo.5-inch  beams  were  much  shorter,  and  the  long  column  influ- 
ence less  marked  ;  consequently  the  values  of  K  are  correspond- 
ingly higher.  The  tests  are  not  sufficiently  numerous  to  fix 
the  law  of  the  decrease  of  .AT  with  the  increase  of  span. 

Beams  Nos.  I  and  2  weighed  200  pounds  per  yard,  with  a 
moment  of  inertia  (/)  equal  to  706.6.  ,  Beam  No.  3  weighed 
105  pounds  per  yard,  and  gave  1=  174.75;  while  No.  4 
weighed  92  pounds  per  yard,  with  /  =  154.9. 


Art.  66.— Built  Flange  Beams  with  Equal  Flanges.— Cover  Plates. 

A  "  built  beam  "  is  a  beam  built  up  of  plates  and  angles 
like  that  shown  in  Fig.  I.  As  shown  in  that  figure  the  web  is 
composed  of  a  single  plate,  called  the  "  web  plate/'  supported 
by  "  stiffeners,"  if  necessary,  as  is  usually  the  case.  These  stiff- 
eners  are  vertical  pieces  of  LS  or  J^s  riveted  to  the  web  plate, 
in  accordance  with  principles  to  be  shown  hereafter.  The 
flanges,  as  shown  by  the  heavy  lines,  are  composed  of  Ls  and 
plates  so  arranged  as  to  give  the  requisite  area  of  cross  section 
at  any  point. 

The  method  of  designing  such  a  beam,  and  the  calculation 
of  the  elements  of  its  resistance,  will  be  given  in  detail.  The 
beam  is  supposed  to  be  of  wrought  iron,  and  one  of  a  system 
for  a  double  track  railway  bridge  ;  the  stringers  under  the  two 
tracks,  which  rest  on  the  beam,  are  placed  at  A  and  B,  and  D 
and  H.  The  weight  of  the  beam,  taken  uniformly  distributed, 
is  5.,3OO.oo  pounds.  The  concentrated  load  at  each  of  the 
points  A,  B,  D  and  H,  composed  of  the  train  weight  added  toj 
that  of  the  stringers,  is  42,000.00  pounds. 


Art.  66.] 


DATA. 


579 


The  following  are  some  of  the  dimensions  of  the  beam: 

Span  RR'  —  26.5  feet.  Depth  of  web  plate  =  36  inches. 
R'H  =  RA  =  3.25  feet.  DH  -  AB  =  6.00  feet.  BD  =  8.00 
feet. 

The  web  plate  will  be  taken  T7ff  inch  thick.  The  method  of 
determining  this  thickness  will  be  shown  hereafter. 

In  this  case  resistance  to  flexure  of  the  web  will  be  neg- 
lected;  the  web  will  be  assumed  to  resist  the  shear  only,  as  is 
assumed  in  Eqs.  (7)  and  (/a)  of  Art.  65.  The  "  depth,"  dv  of  the 


•  1 

If 

A                             B 

'    '1 

1  

C 

• 

: 

RJ 

i 
1 

Fig.1 

i 

beam  will  then  be  the  vertical  distance  between  the  centres  of 
gravity  of  the  sections  of  the  flanges,  and  each  flange  is  to  be 
considered  as  composed  of  two  LS  and  the  "  cover"  plate  or 
plates  only  ;  no  part  of  the  web  is  to  be  included.  Strictly 
speaking,  then,  the  depth  is  variable  ;  but  this  variation  is  so 
slight  that  no  essential  error  will  be  committed  if  it  be  con- 
sidered constant  and  equal  to  the  depth  of  web  plate,  or  36 
inches.  This  procedure,  which  saves  much  labor  and  time,  is 
always  permissible  where  cover  plates  are  used.  The  next  ex- 
ample will  exhibit  a  case  in  which  they  are  not  used. 

The  direct  stresses  of  tension  and  compression  existing  in  the 
flanges  must  be  carried  through  the  rivets  which  unite  the  flanges 
to  the  web  ;  hence  the  necessary  number  of  those  rivets  will 
first  be  determined. 

The  reaction  at  R,  using  the  data  already  given,  will  be  : 


R  =  2  x  42,000  oo  + 


E      ?  =  86,650.00  pounds. 


580  BUILT  BEAMS.  [Art.  66. 

The  weight  per  lineal  foot  of  floor  beam  is : 

5,300.00 

^— =  200.00  pounds  =  w. 

26.5 

The  bending  moments  for  the  two  sections  A  and  By  will 
next  be  found. 


Moment  at 
A  =  (86,650  —  100  X  3.25)  3.25  =  280,500.00  nearly. 

Moment  at 

•. 

B  =  86,650  X  9.25  —  42,000  X  6  —  IOD  (9.2 5)2  =  463,950.00. 

Since  the  depth  of  the  beam  is  3  feet : 
Flange  stress  at 

A  =  280,500.00  '-f-  3  =  93,500.00  pounds. 
Flange  stress  at 

B  =  463,950.00  -r-  3  =  1 54,650.00  pounds. 

The  allowable  intensity  of  pressure  between  the  rivet  and 
its  hole  (see  Art.  73)  will  be  taken  at  10,000.00  pounds.  The 
diameter  of  rivets  is  a  matter  of  judgment ;  it  will  be  taken  at 
%-inch.  Rivets  for  built  beams  usually  range  from  ^  to  I 
inch  in  diameter. 

The  selection  of  the  Ls  for  the  flanges  is  also,  to  some 
extent,  a  matter  of  judgment.  In  the  present  instance, 
4"  X  4"LS>  5°  pounds  per  yard,  will  be  taken.  These  will  be 
found  to  answer  the  purpose. 


Art.  66.]  RIVETS  AND  FLANGE   STRESSES.  581 

The  effective  bearing  surface  between  each  rivet  and  the 
web  plate  will  then  be  : 

|  X   -4  =  0.383  square  inch. 

5  IO 

Hence  each  rivet  may  carry: 

0.383   X   10,000.00  =  3,830.00  pounds. 
Consequently  the  number  of  rivets  between  R  and  A  should  be  : 

93,500.00  -T-  3,830.00  =  24  (nearly). 
The  increase  of  flange  stress  between  A  and  B  is: 

154,650  —  93,500  =  61,150.00  pounds. 

Hence  the  number  of  rivets  required  between  A  and  B  is  : 
61,150.00  -5-  3,830.00  =  16  (nearly). 


. 

Since  24  rivejs  are  required  between  R  and  A,  the  corre- 
sponding pitch  would  be  but  a  little  more  than  one  and  one- 
half  inches/  which  is  very  much  too  small.  With  a  /6-inch 
rivet,  a  three-inch  pitch  is  about  the  least  advisable.  If  the 
rivets  be  placed  at  a  pitch  of  three  inches  between  R  and  B, 
thirty-seven  will  thus  be  located.  Three  or  four  rivets,  if  nec- 
essary, may  be  put  in  at  the  end  with  a  pitch  of  two  inches 
without  harm.  Again,  plates  at  the  upper  corners  of  the  beam, 
as  shown,  carrying  at  least  six  rivets  each,  should  be  put  on. 
In  such  methods  as  these,  nearly  the'  full  number  of  rivets 
required  between  R  and  A  may  be  supplied,  while  the  two  or 
three  lacking  will  be  found,  without  danger  to  the  b^m,  adja- 
cent  to  A  on  the  side  towards  B.  Three  or  four  in  excess  of 
the  number  required  will  be  found  between  A  and  B. 


582  BUILT  BEAMS.  [Art.  66. 

No  central  bending  moment  at  C  has  been  computed,  be- 
cause the  only  difference  between  such  a  one  and  that  at  either 
B  or  D  is  due  to  the  weight  of  the  beam  only.  This  difference 
is  essentially  nothing.  The  proper  support  of  the  Ls  *n  com- 
pression, however,  requires  that  the  rivets  be  pitched  at  about 
six  inches  between  B  and  D.  In  ordinary  floor  beams  a  proper 
bond  between  the  flanges  and  web  requires  that  the  pitch  should 
never  be  greater  than  about  six  or  eight  inches. 

The  shearing  of  the  rivets  is  not  considered,  because  they 
sustain  double  shear  in  the  flanges,  and  their  bearing  capacity 
is  by  far  the  least  of  the  two. 

The  rivets,  of  course,  should  be  pitched  alike  in  both  top 
and  bottom  flanges. 

The  greatest  allowable  intensity  of  tensile  stress  in  the  bot- 
tom flange  will  be  taken  at  8,000.00  pounds  per  square  inch, 
and  an  equal  intensity  will  be  taken  for  the  compressive  stress 
in  the  upper  flange.  The  area  required  in  the  bottom  flange 
at  A  is  : 

=  11.7  sq.  ins.  (nearly). 


8,000 
That  required  at  B  is  : 

154,650  ,  N 

3^000      ~-  J9-3  sq.  ms.  (nearly). 

The  area  of  the  two  4"  X  4"  LS,  50  pounds  per  yard,  is  10.00 
square  inches.  The  thickness  of  the  angle  iron  where  it  is 
pierced  by  the  rivets  binding  it  to  the  web  is  about  O.6  inch. 
Hence  the  area  of  metal  taken  out  by  one  rivet  is  : 

0.875  X  0.6  x  2  =  1.05  sq.  ins. 

Or,  the  effective  area  of  the  LS  at  A  is  : 

,10.00  —  1.05  =  8.95  square  inches. 


Art.  66.]  FLANGES.  583 

Now,  since  the  weight  of  the  beam  itself  is  small,  compared 
with  the  weight  of  the  train,  the  flange  stress,  or  moment, 
varies  almost  uniformly  from  R  to  A.  Hence,  an  increased 
section  is  first  needed  at  V*  •  ?  3  .  2.  5*  •  '  fr%  ?  $"  »  /  /,  7 

(8.95  -T-  11.7)  X  3.25  =  2.5  feet  (nearly), 

from  R.  Since,  however,  the  cover  plate  to  be  added  must 
take  its  stress  through  the  rivets  which  bind  it  to  the  LS,  it 
should  overlap  the  necessary  distance  by  one  and  a  half  to 
twice  its  width.  In  the  present  case,  then,  instead  of  begin- 
ning the  cover  plate  at  just  2.5  feet  from  R,  a  12"  x  iV"  cover 
plate  will  begin  at  9  inches  from  R  and  extend  along  the  beam 
to  a  point  at  the  same  distance  from  R.  The  length  of  this 
cover  plate  will  then  be  26.5  —  1.5  =  25  feet.  This  cover  plate 
will  be  bound  to  the  angle  irons  by  ^"  rivets,  which  should,  so 
far  as  possible,  be  pitched  half  way  between  the  J/s"  rivets  in 
the  other  legs  of  the  angle  irons.  The  effective  area  of  this 
cover  plate,  for  tensile  stress,  will  then  be  : 

(12  --  4erf^  X  A  =  5-9  sq.  ins.  (nearly). 


The  available  area  of  two  LS  and  one  cover  plate  is,  conse- 
quently : 

8-95  +  5-9  =  I4-85  sq.  ins. 

For  the  reason  already  given,  the  moment,  or  flange  stress, 
varies  nearly  uniformly  between^!  and  B,  but  at  a  different  rate 
than  between  R  and  A.  Since  AB  is  6.00  feet,  the  point  at 
which  another  increase  of  section  must  begin  is  at  the  distance 

[(14.85  —  11.7)  -T-  (19.3  —  11.7)]  X  6.00  =  2.5  feet  (nearly) 

Of.rr*"^)  J(?f"J  7  "*7j  *'  •'  <*  •  ^ 

from  A.    Again,  as  in  the  previous  instance,  a  second  cover 
plate,  12"  X  J4",  will  be  put  on,  and  it  will  begin,  not  at  2.5 


.  I 


// 


584  BUILT  BEAMS.  [Art.  66. 

feet  from  A,  but  at  one  foot  from  that  point.     The  available 
area  of  this  plate  will  be : 

(12  --  1.5)  X  tf  =  5.25  sq.  ins. 

The  total  area  at  the  centre  of  the  beam  available  for  ten- 
sion will  then  be : 

14.85  +  5.25  =  20.10  sq.  ins. 

As  19.3  square  inches,  only,  is  required  at  J5,  the  total  effect- 
ive flange  area  of  20.10  square  inches  will  be  sufficient.  The 
tension  flange  has  been  considered,  but  the  same  design  will 
evidently  give  a  sufficiently  strong  compression  flange.  This 
arises  from  two  causes.  In  the  first  place,  it  has  already  been 
seen  that  the  ultimate  tensile  and  compressive  resistances  of 
wrought  iron  may  be  taken  essentially  the  same.  Again,  in 
first  class  riveted  work  the  rivets  so  thoroughly  fill  the  holes 
that  the  metal  taken  out  by  the  punch  or  drill  need  not  be 
deducted ;  in  other  words,  the  effective  area  is  equal  to  the 
total  area  of  flange  section. 

The  number  of  rivets  required  in  a  cover  plate  is  yet  an 
important  question.  Since  all  stress  carried  by  the  cover  plates 
must  be  given  to  them  by  the  rivets,  the  number  of  rivets. be- 
tween tJie  end  of  any  cover  plate  and  that  point  at  which  a  fur- 
ther increase  of  flange  section  is  necessary,  must  be  sufficient  to 
carry  all  the  stress  in  the  cover  plate  itself. 

Applying  this  principle  to  the  first  cover  plate  found  neces- 
sary^ The  load  which  each  £"  rivet  in  the  12"  x  -ft-"  cover  may 
carry  is : 

°-75  X  A  X  10,000.00  =  4,220.00  pounds. 

The  total  tensile  stress  carried  by  the  12"  x  -£?"  cover  is  : 
5.9  X  8,000  =  53,200.00  pounds.  Hence  the  number  of  rivets 
required  is : 


Art.  66.]  FLANGES.  585 

53,200  -T-  4,220  =  13  (nearly). 

According  to  the  design  it  is  5  feet  from  the  end  of  this 
cover  to  a  point  2.5  feet  from  A  toward  B,  where  the  next  in- 
crease in  section  is  required  ;  and  over  this  5  feet  these  13 
rivets  must  be  distributed.  But  in  order  that  a  proper  bond 
between  the  component  parts  of  the  flange  may  be  obtained,  it 
it  seldom  advisable  to  make  the  pitch  over  6",  and  at  the  end 
of  the  cover  plate  this  pitch  should  be  halved  for  about  three 


Fig.2 


rivets.  Proceeding  in  this  manner,  that  part  of  the  bottom  of 
the  beam,  at  the  end  nearest  R  in  Fig.  I,  which  includes  the  5 
feet  of  cover  under  consideration,  will  present  the  appearance 
of  the  sketch  in  Fig.  2.  RG  is  0.75  foot  and  GF  5  feet.  In 
this  manner  22  rivets  are  introduced  instead  of  13,  but  it  is 
advisable  to  put  in  the  extra  number. 

In  the  compression  flange  other  considerations  appear  be- 
sides the  simple  bearing  capacity  of  the  shaft  of  the  rivet. 
Between  any  two  consecutive  rivets  the  cover  plate  forms  a  solid 
rectangular  column  with  essentially  fixed  ends,  whose  length  is  the 
pitch  of  rivets.  The  pitch,  therefore,  must  not  be  sufficiently 
great  to  allow  the  existence  of  any  material  amount  of  long 
column  flexure.  Unless  plates,  therefore,  are  very  heavy,  the 
greatest  pitch  should  not  exceed  about  six  or  eight  inches. 

The  bearing  capacity  of  a  }"  rivet  in  12"  x  \"  cover  is  : 

0.75  X  0.5   X   10,000.00  =  3,750.00  pounds. 
The  full  tensile  capacity  of  the  cover  plate  is  : 


586  BUILT  BEAMS.  [Art.  66. 

5.25  X  8,000.00  —  42,000.00  pounds. 
Hence,  the  number  of  rivets  required  is  : 

42,000.00  -r-  3,750.00  =   1 1  (nearly). 

The  end  of  the  cover  plate,  as  designed,  is  one  foot  from  A 
towards  B  ;  arid  the  eleven  rivets  are  nearly  all  required  be- 
tween that  end  and  B,  a  distance  of  5  feet.  Hence,  if  the 
rivets  are  pitched  in  this  cover  plate,  near  the  ends,  as  shown 
in  Fig.  2  for  the  other  cover,  and  at  six  inches  over  the  inter- 
vening space,  more  than  the  number  just  determined  will  be 
introduced.  For  the  reasons  already  given,  however,  the  num- 
ber will  really  be  not  too  great. 

In  each  flange,  then,  there  will  be  found  the  following 
pieces  properly  joined : 

2 —  4"  x    4"  LS,  50  pounds  per  yard. 
I  — 12"  X  Ty  plate,  25  feet  long. 
I — 12"  X    i"  plate,  18  feet  long. 

At  the  ends  of  the  beam  R  and  R',  Fig.  i,  provision  must 
be  made  for  the  reaction.  In  this  example  the  reaction  is 
86,650.00  pounds.  The  transverse  shearing  resistance  of  the 
web  should  at  least  equal  this  at  the  ends.  The  area  of  a  trans- 
verse section  of  the  web  is  : 

36  X  TV  =  I575  sq.  ins. 

If  the  greatest  allowable  shearing  intensity  in  the  web  be 
taken  at  8,000.00  pounds,  its  shearing  resistance  will  be : 

:5-75  X  8,000.00  =  126,000.00  pounds. 
This  result  is  about   50  per  cent,   greater  than  is  required. 


Art.  66.]  ENDS.  587 

Hence  safety,  so  far  as  shearing  is  concerned,  is  amply  secured. 
But  the  end  of  the  beam  is  also  subject  to  an  upward  pressure 
of  86,650.00  pounds,  which  must  also  be  provided  for.  Two 
6"  X  4"  X  i"  LS  will  be  riveted  to  the  ends  as  shown  in  Fig. 
I,  one  on  each  side  of  the  web,  and  the  6"  legs  lying  against 
it.  By  pitching  J"  riyets  at  3"  (nearly),  in  a  zigzag  manner, 
20  rivets  can  be  introduced  to  hold  these  4"  x  6"  x  J"  LS  to 
the  web.  The  carrying  capacity  of  each  f"  rivet  against  the 
web  plate  has  already  been  found  to  be  3,830.00  pounds. 
These  20  rivets  therefore  will  carry  3,830  X  20  =  76,600.00 
pounds.  Since  the  area  of  the  cross  section  of  two  4"  X  6"  x 
$•"  Ls  is  about  10  sq.  ins.,  the  bearing  of  the  rivets  against  the 
web  plate  is  all  that  need  be  considered  in  this  connection. 

A  proper  bearing  for  the  difference  86,650  —  76,600  = 
10,050.00  pounds  remains  to  be  found.  If  it  be  supposed  that 
the  ends  of  the  beam  rest  on  "shoes"  or  brackets,  or  other 
supports,  it  is  only  necessary  that  at  least  2\  inches  of  the  edge 
of  the  web  plate  bear  on  such  support ;  for  the  bearing  surface 
will  be  2j  X  TV  =  i.oo  sq.  in.  (nearly),  and  it  will  carry 
10,050.00  pounds.  In  all  ordinary  cases  much  more  than  that 
amount  of  edge  will  bear  on  the  support.  It  is  to  be  remem- 
bered that  in  such  an  instance  as  this,  the  lower  ends  of  the  4" 
X  6"  LS  must  bear  fairly  and  truly  against  the  angle  irons  com- 
posing  tJie  lower  flange,  in  order  that  they  may  take  up  their 
proper  amount  of  the  reaction. 

In  some  cases  the  ends  of  the  beam  are  to  be  secured  to 
vertical  surfaces  without  any  supporting  shoe  or  bracket.  The 
entire  reaction  of  such  a  beam  must  be  carried  by  the  vertical 
angles  at  the  ends.  The  number  of  -J"  rivets  required  to  hold 
these  angle  irons  to  the  web  would  then  be  86,650  -r-  3,830  = 
23  (nearly).  By  shortening  the  pitch  a  little  these  could  easily 
be  worked  into  the  longer  legs  of  the  6"  X  4"  X  £"  Ls-  12 
rivets  would  then  be  put  through  each  of  the  two  4"  legs  and 
the  vertical  surface  to  which  the  beam  is  secured. 

No  account  has  heretofore  been  taken  of  the  shearing  re- 


BUILT  BEAMS.  [Art.  66. 


sistance  of  the  rivets,  because  that  has  been  much  greater  than 
their  bearing  capacity,  but  instances  may  occur  in  which  such 
a  condition  of  things  does  not  exist.  Hence  the  shearing 
and  bearing  capacities  should  always  be  estimated,  and  se- 
curity taken  in  reference  to  that  which  is  least.  As  an  ex- 
ample :  at  8,000.00  pounds  per  sq.  in.  the  shearing  resistance 
of  a  |"  rivet  is  (o.S/S)2  X  0.7854  X  8,000.00  =  4,800.00  pounds 
(nearly)  ;  while  the  bearing  capacity  of  the  same  rivet  in  the 
6"  X  4"  X  i"  L  is  only  : 

0.875  X  0.5  x  10,000.00  =  4,375.00  pounds. 

Precisely  the  same  operations  are  required  in  determining 
the  number  of  rivets  in  the  vertical  l^s  at  A  and  B,  Fig.  I,  as 
in  those  at  the  ends  of  the  beam  ;  consequently  it  is  not  neces- 
sary to  repeat  them. 

Thus,  there  is  completed  the  operation  of  designing  the 
beam,  with  the  exception  of  finding  the  thickness  of  the  web, 
which  will  be  given  hereafter. 

In   general  two  or  three  things  are  to  be  observed.      The 

I  number  of  rivets  actually  required  by  these  calculations  should  al- 

I  ways  be,  as  they  just  have  been,  somewhat  exceeded.     In  the  best 

\of  riveted  work  the  rivets  will  not  exactly  fill  the  holes,  and 

the  beam  will  not  act  perfectly  as  one  continuous  whole. 

Again,  stress  is  given  to  the  flanges  along  the  line  of  the 
rivet  holes,  which  is  some  distance  from  the  centre  of  gravity 
of  the  cross  section  of  the  flange.  Consequently,  some  bend- 
ing will  be  induced  in  both  flanges,  and  this  necessitates  some 
extra  material.  This  excess  may  be  estimated  if  desirable,  but 
ordinarily  it  is  entirely  unnecessary.  The  existence  of  this 
bending  demonstrates  the  advisability  of  putting  on  as  few 
cover  plates  as  possible.  It  is  far  better  to  use  heavier  Ls  with 
a  little  waste  of  material  at  the  ends. 

It  is  also  better  to  use  one  heavy  cover  plate  than  two  thin 
ones  having  an  equal  combined  thickness,  even  though  the  use 


Art.  67.] 


DATA. 


589 


of  the  former  entails  a  little  waste  ;  for  the  heavy  plate  be- 
tween two  consecutive  rivets  will  resist  far  more  bending  as  a 
column  than  the  two  others  each  of  half  the  thickness. 

If  the  end  of  the  beam  were  made  as  shown  in  Fig.  3,  no 
web  plate  would  be  re- 
quired between  R  and 
Ay  for  all  shear  would 
be  carried  by  the  in- 
clined flange. 

The    upper  flange, 
being  in  compression, 

would  require  riveting,  but  none  would  be  needed  in  the 
lower,  except  in  the  immediate  vicinity  of  R.  The  flange 
stresses  between  A  and  R  would  also  be  uniform,  instead  of 
uniformly  varying  as  in  Fig.  I . 


'  Fig.3 


Art.  67.— Built  Flanged  Beams  with  Equal  Flanges.— No  Cover  Plates. 

The  flanged  beam  represented  in  Fig.  I  is  supposed  to  carry 
a  portion  of  the  floor  of  a  highway  bridge^  In  this  case,  also, 
the  bending  resistance  of  the  web  plate  will  be  neglected.  The 
beam  proper  is  the  portion  RR'  R'Ry  supported  at  RR  and 
R'R  ;  while  the  portions  ARR  and  HR'R  form  cantilevers  for 
the  support  of  the  sidewalks. 

The  following  are  the  dimensions  : 

AR  =  HR  =    6  feet.  RR'  =  28  feet. 

AH  =  40  feet.          RR  =  R'R'  =  31  inches. 
RB  =  BM  =  MF  =  FR  =  7  feet. 


The  depth  RR  has  been  taken  at  31  inches,  so  that  the 
effective  depth  to  be  used  in  finding  the  flange  stresses  will 
be  about  2.5  feet. 


5QO 


BUILT  BEAMS. 


[Art.  67. 


The  weight  of  the  beam   proper,   RR'R'R,   added   to  the 

flooring  which  it  supports,  is  taken  at 

/  1 


14,650.00  pounds. 


at 


The  greatest  uniform  load  between  R  and  R'  will  be  taken 


37,440.00  pounds. 


M 


C  I  D 

Fig,1 


Hence  the  total  uniform  load  to  which  the  beam  is  sub- 
jected is : 

37,440.00  +  14,650.00  —  52,090.00  pounds. 

The  weight  of  one  cantilever,  with  the  flooring  which  it  sup- 
ports, will  be  taken  at 

3,100.00  pounds. 
The  total  moving  load  on  AR,  or  HR',  will  be  taken  at 

8,640.00  pounds. 
The  total  load,  therefore,  carried  by  one  cantilever  is  : 

3,100.00  -f  8,640.00  =  11,740.00  pounds. 
The  beam  proper,  RR',  may  sustain  its  greatest  load  when 


Art.  67.]  NO   COVER  PLATES.  591 

the  sidewalks  carry  nothing  but  their  fixed  weight.  This  con- 
dition of  things  will  cause  the  greatest  compression  in  the 
upper  flange  ,and  tension  in  the  lower,  and  will  be  assumed  in 
designing  the  beam. 

The  fixed  weight  of  a  cantilever  will  cause  stresses  in  the 
flanges  of  opposite  kinds  to  those  produced  in  the  beam,  but 
of  such  small  amount  that  they  will  be  neglected  ;  the  neglect 
originating  a  very  small  error  on  the  side  of  safety. 

The  total  load  per  linear  foot  of  RR  is  : 

52,090.00  -4-  28  =  1,860.00  pounds. 

The  flange  stress  in  the  beam  at  R  will  be  nothing  ;  it  will 
be  found  at  the  two  points  B  and  M.  Strictly,  the  "  depth  " 
to  be  used  should  be  the  vertical  distance  between  the  centres 
of  gravity  of  the  flanges.  It  will  not  be  far  wrong  to  take  this 
depth  at  2.5  feet,  since  the  web  plate  is  31  inches  deep.  The 
reaction  at  R  is : 

52,090.00  -4-  2  =  26,045.00  pounds. 
The  flange  stress  at  B  is : 
(26,045  x  7  —  i, 860  X  (7)*  -7-  2)  -7-  2.5  =  54,700.00  pounds. 

The  flange  stress  at  the  centre  M  is  : 

• 
(52,090  X  28  -r-  8)  -f-  2.  5  =  72,926.00  pounds. 

If,  as  in  the  preceding  Article,  the  greatest  allowable  stress 
in  the  flanges  is  8,000.00  pounds  per  square  inch,  a  flange  area 
of  9.115  square  inches  is  required  in  the  present  case.  If  each 
flange  is  composed  of  2 — 4"  x  6"  x  %"  LS,  50  pounds  per 
yard,  there  will  be  a  very  little  excess  of  flange  area,  as  there 


592  BUILT  BEAMS.  [Art.  67. 

should  be  ;  these  LS  will  then  be  taken  for  the  flange,  the  4" 
legs  being  riveted  to  the  web  plate  ;  -J"  rivets  will  be  used  ifi 
riveting  the  flanges  to  the  web.  Where  pierced  by  the  rivets, 
the  legs  of  the  Ls  are  about  |-"  thick.  Hence  a  rivet  hole  will 
cut  out  2  X  i  X  0.875  =  0.875  square  inch.  There  will  then 
still  remain  10.0  —  0.875  =  9.125  square  inches  of  effective 
area,  which  is  a  little  in  excess  of  the  9.115  required. 

A  web  plate  f"  thick  will  be  assumed.  Taking  10,000.00 
pounds  per  square  inch  as  the  greatest  allowable  intensity  of 
pressure  between  shaft  of  rivet  and  plate,  the  bearing  capacity 
of  each  rivet  will  be : 

0.875  X  0.375   X   10,000  =  3,280.00  pounds. 

In  this  case  all  the  moving  load  rests  upon  the  top  of  the 
beam,  and  since  the  edge  of  the  web  plate  is  only  0.375"  wide, 
that  moving  load  must  be  taken  as  resting  on  the  LS  of  the 
upper  flange,  and  hence  indirectly  on  the  rivets.  Also,  since 
nearly  the  whole  of  the  fixed  load  rests  upon  the  upper  flange, 
the  entire  load  of  the  beam  will  be  taken  as  resting  on  that  flange. 
Consequently,  between  R  and  B  the  rivets  will  be  subjected  to 
the  action  of  a  vertical  force  equal  to  1,860  X  7  =  13,020.00 
pounds,  and  a  horizontal  one  equal  to  54,700.00  pounds.  The 
resultant  force  will  then  be  : 


V(i3>02o)2  +  (54,700)"  —  56,230.00  pounds. 

Between  B  and  M  the  vertical  force  will  then  be  the  same, 
but  the  horizontal  one  will  be 

72,926.00  —  54,700.00  =  18,226.00  pounds. 

The  resultant,  therefore,  is : 

^ __ 

<V/(i3,02o)a-f  (i8,226)a  =  22,400  pounds. 


Art.  67.]  NO   COVER  PLATES.  593 

Hence  the  number  of  rivets  required  between  R  and  B  is : 

56,230.00  -r-  3,280.00  =  1 8  (nearly). 
The  number  between  B  and  M  is  : 

22,400.00  -f-  3,280.00  =  7  (nearly). 

If,  therefore,  commencing  at  R  or  R',  the  rivets  be  pitched 
at  3  inches  for  a  distance  of  4.5  feet,  then  at  6  inches  to  the 
centre  M,  about  36  or  37  rivets  will  be  found  in  each  half  of 
each  flange.  This  number  is  in  excess  of  that  required,  but 
for  the  reasons  given  in  the  preceding  Article,  it  is  probably 
not  too  many.  Thus  the  flanges  are  designed  without  the  use 
of  cover  plates. 

In  this  case  the  beam  will  be  suspended  from  hanger  loops 
at  R  and  R',  which  carry  resting  plates  or  shoes  for  the  beam 
at  their  lower  extremities. 

The  total  reactions  at  the  lower  R  and  R'  will  be  half  the 
total  weight  of  the  entire  beam  with  the  moving  load,  or : 

Reaction  =  (52,090.00  +'23,480.00)  -f-  2  —  37,785.00  pounds. 

At  R  and  R  2—4"  X  4"  X  i"  LS  will  be  riveted  to  the  beam 
as  shown.  The  lower  ends  of  these  angles  should  abdt  firmly 
and  squarely  against  the  angles  of  the  lower  flange. 

Since  the  greatest  allowable  pressure  between  a  rivet  and 
the  web  plate  is  3,280  pounds,  the  number  of  rivets  required 
at  each  end  of  the  beam  in  each  pair  of  vertical  LS  is : 

37,785.00  -^  3,280.00  =  12  (nearly). 

If,  consequently,  these  rivets  be  pitched  at  3  inches,  a 
sufficient  number  will  be  obtained,  if  it  be  remembered  that  a 


594  BUILT  BEAMS.  [Art.  67. 

portion  of  the  edge  of  the  web  will  bear  against  the  resting 
plate. 

The  pitch  in  the  stiffners  (4"  X  4"  X  •&"  LS)  at  C  and  D 
may  be  taken  at  6"  with  an  extra  rivet  at  each  end. 

The  horizontal  flange  stress  for  the  cantilevers  at  R  and  R 
is : 

(11,740.00  X  3)  -T-  2.5  —  14,088.00  pounds. 

The  secant  of  the  angle  which  the  inclined  flange  makes 
with  the  horizontal  is  about  1.05.  Hence  the  inclined  flange 
stress  is  : 

14,088  X   1.05  =  14,800.00  pounds. 
Hence,  if  in  each  flange  at  A  there  are 

14,800.00  -r-  3,280  =  5  (nearly) 

rivets,  securing  the  flanges  to  the  piece  of  plate  shown,  ample 
security  will  be  obtained. 

The  cantilever  flanges  possess  a  large  excess  of  material. 

Calculations  on  the  shearing  of  the  rivets  between  the  web 
and  flange  have  not  been  made,  because  the  resistance  of  a 
rivet  to  double  shear  is  much  in  excess  of  its  bearing  capacity. 

The  excess  of  material  in  the  LS  of  the  flanges  is  not  as 
much  as  it  really  should  be,  because  the  line  of  horizontal 
stress  along  the  rivet  holes  is  somewhat  below  or  above  the 
centre  of  gravity  of  the  flange,  and  some  bending  is  conse- 
quently induced.  This  bending,  however,  is  not  as  great  as  if 
cover  plates  had  been  used,  and  the  neglect  of  the  bending 
resistance  of  the  web  plate  is  somewhat  of  an  offset.  Besides, 
as  has  already  been  stated,  in  this  particular  case,  the  fixed 
weight  of  the  cantilevers  relieves  a  little  of  the  flange  stress  of 
the  beam  as  actually  found. 

Since  the  transverse  section  of  the  web  plate  has  an  area  of 


Art.  68.] 


BOX  BEAMS. 


595 


°-375  X  3°  —  11-25  square  inches,  transverse  shearing  at  the 
points  of  support  is  more  than  provided  for. 

If  either  a  railway  or  highway  floor  beam  has  a  variable 
depth,  the  operations  are  in  no  manner  changed.  The  depth, 
however,  to  be  used  in  finding  the  flange  stress  at  any  point 
must  be  the  vertical  depth  at  that  point.  The  stress  thus 
determined  must  be  multiplied  by  the  secant  of  the  inclination 
to  the  horizontal  at  the  same  point  for  the  inclined  flange. 

Art.  68. — Box  Beams. 

The  class  of  beams  known  as  box  beams  in  engineering 
practice  are  represented  in  Figs.  I  and  2.  In  Fig.  I  the  upper 
and  lower  flanges  are  each  composed  of  a  plate  whose  thick- 
ness is  /'  and  two  Ls  whose  lengths  of  legs  and  thickness  are 
s  and  <?,  respectively.  If  it  be  assumed  that  the  web  plates, 


* & 


Fig.1  ! 


Fig.2 


Fig.3 


the  thickness  of  each  of  which  is  /,  offer  no  resistance  to  bend- 
ing, then  the  effective  depth  of  the  box  beam  will  be  the  ver- 
tical distance  between  the  centres  of  gravity  of  the  flanges. 
If  /  is  the  area  of  one  of  these  flanges,  and  dt  this  effective 
depth,  the  resisting  moment  of  the  beam,  as  has  already  been 
shown,  will  be  : 


M=C/dl; (I) 


FORMULA  FOR  BUILT  BEAMS.  [Art.  69. 

in  which  C  —  K  =  intensity  of  stress  at  the  distance  y2d^  from 
the  neutral  surface.     If  the  flange  area  is  desired  : 


« 


In  other  words,  the  methods  and  all  the  operations  regard- 
ing rivets,  etc.,  as  well  as  the  values  of  C  and  7",  or  K,  are 
precisely  the  same  for  the  box  beams  as  for  the  other  built 
beams  of  the  preceding  Articles. 

If  each  flange  is  composed  of  several  plates  and  4  |_s  (as 
shown  by  one  in  dotted  lines),  then  /'  is  to  be  taken  as  the 
combined  thickness  of  all  the  plates,  while  f  will  be  the  com- 
bined area  of  the  several  plates  and  4  LS. 

Fig.  2  shows  a  box  beam  composed  of  two  channels  and 
one  or  more  plates  in  each  flange.  The  general  observations 
applied  to  Fig.  I  apply  with  equal  force  to  Fig.  2.  The  bend- 
ing resistance  of  the  webs  of  the  channels  may  be  neglected  if 
very  thin,  or  when  desired  in  any  case,  but  the  exact  formula 
to  be  given  in  the  next  Article  is  well  adapted  to  this  beam. 


Art.  69. — Exact  Formulae  for  Built  Beams. 

The  exact  formulae  for  the  built  sections  already  given  are 
simply  the  special  forms  of  the  general  formula  : 


The  moment  of  inertia,  7,  is  to  be  taken  about  a  horizontal 
line  through  the  centre  of  gravity  of  the  normal  section,  i.e., 
about  a  line  parallel  to  the  side  b  in  the  three  Figs,  of  the  pre- 
ceding Article. 


Art.  69.]  FORMULAE.  597 

In  Fig.  i  of  that  Article  the  moment  of  inertia  of  the  cross 
section  about  AB  is: 


, 


)n 


If  there  are  four  Ls  m  each  flange,  one  .only  of  which  is 
shown  in  dotted  lines: 


r=  i    t  i 

62 


_  r 


The  moment  of  inertia  of  the  cross  section  shown  in  Fig.  2, 
about  AB,  is  : 


The  moment  of  inertia  of  the  cross  section  shown  in  Fig. 
3,  about  AB,  can  be  either  directly  written  from  an  examina- 
tion of  that  Fig.,  or  derived  from  Eq.  (2)  by  simply  writing 


—  for  t.     It  has  the  value  : 
2 


n>  -a)(d-  2a)3  +  a(d  -  2/T]  , 

6  •.;;••;•••  W 


598  BUILT  BEAMS.  [Art.  70. 

If  the  plates  are  omitted  from  the  flanges  in  Fig.  3,  as  in 
the  Article  on  built  beams  without  cover  plates,  t'  =  o,  and 


_  (s  +  tftyl*        r(s  -a)(d-  2*  )3  -f  a(d  - 
~~~  ~ 


r(s  -a)(d-  2*  )3  -f  a(d  -  2Syr\ 
^6~ 


_ 


In  all  these  cases  </,  =.  y^d  '  +  t'  or  y^d,  according  as  there 
are  or  are  not  cover  plates. 

These  several  .values  of  /and  d»  substituted  in  Eq.  (i),  will 
give  the  resisting  moments  for  the  various  sections.  It  is  an 
open  question,  however,  what  degree  of  accuracy  may  be  ex- 
pected to  result  in  the  application  of  these  formulae.  It  is  to 
be  remembered  that  the  very  best  of  riveted  work  does  not  se- 
cure that  degree  of  continuity  presupposed  by  the  Eq.  (i).  It 
may  be  stated,  however,  that  Eq.  (4)  is  better  applicable  to  its 
cross  section  than  the  others,  for  there  is  perfect  continuity  be- 
tween the  web  and  a  part  of  the  flange. 


Art.  70.— Examples  of  Built  Beams  Broken  by  Centre  Weight. 
Example  I. —  Wrought  Iron  Beam. 

This  beam  was  tested  by  Sir  William  Fairbairn  (u  Useful 
Information  for  Engineers,"  first  series),  and  was  composed  of 
four  2-inch  LS  riveted  to  a  7  by  ^-inch  web  plate.  The  dis- 
tance between  supports  was  7  feet  or  84  inches. 

A  section  of  the  beam  is  shown  by  the  section,  only,  of  Fig. 
i  in  Art.  67 ;  there  were  no  cover  plates. 

The  LS  in  the  bottom  flange  were  a  very  little  heavier  than 
those  in  the  upper,  but  the  difference  was  so  small  that  it  has 
been  neglected  ;  or,  rather,  the  small  excess  has  been  assumed 
to  supply  the  loss  caused  by  the  rivet  holes. 

Centre  breaking  weight  —  24,380  +  80  =  24,460  Ibs. 


Art.  70.]  BREAKING    TESTS.  599 

Wl 

1=7  feet  =  84  inches.     /.     M  =  -   -  =  513,660. 

4 

Referring  to  Eq.  (6)  of  Art.  65  : 

d  —  6.5  ins.        /=  2.083.         -4  =  0.30. 


=  '5.5. 

=  33,140  Ibs.  per  sq.  in. 


d^  was  taken  as  the  depth  (nearly)  between  the  roots  of  the  LS. 

The  beam  gave  way  in  the  top,  or  compression,  flange  by 
the  twisting  of  the  LS  at  a  comparatively  low  compressive  in- 
tensity. This  indicates  that  the  discontinuous  riveted  connec- 
tion between  the  web  and  flange,  although  the  pitch  of  the 
rivets  was  only  4.5  inches,  fails  to  give  such  perfect  support  to 
the  top  flange,  as  a  column,  as  the  perfect  continuity  of  the 
connection  in  a  rolled  beam. 

The  condition  of  the  top  flange,  as  a  column,  in  a  built 
beam,  therefore,  exercises  a  very  important  influence  on  the 
ultimate  resistance  of  the  beam,  and  should  not  be  neglected. 

It  is  probable,  however,  that  the  high  compressive  resist- 
ance of  American  wrought  iron  of  the  present  day  would  give 
a  much  higher  value  of  C  under  the  same  circumstances. 

When  the  centre  load  added  to  five-eighths  the  weight  of 
the  beam  was  8,400  pounds,  the  centre  deflection,  or  w,  was 
O.I  8  inch.  Hence  the  coefficient  of  elasticity  was: 


—  12,321,000  pounds. 


/3  must  be  taken  in  inches.     /  was  computed  by  Fairbairn 
at  46.77. 


6OO  BUILT  BEAMS.  [Art.  70. 

Example  II. — Steel  Beam. 

The  data  for  this  beam  were  given  by  Albert  F.  Hill,  C.  E., 
in  "  Steel  in  Construction,"  Engineers'  Soc.  of  West.  Penn., 
April,  1880.  Each  flange  was  composed  of  two  2^  x  2^  X  -^ 
steel  angles,  and  one  5-j-  X  -ft-  cover  plate.  The  web  was  a 
12  X  T3F"0-5oC"  rolled  steel  plate.  The  clear  span  was  5 
feet;  pitch  of  rivet,  4.5  inches;  total  effective  area  of  section, 
8.51  square  inches.  The  rivet  holes  were  drilled  T\  inch  in 
diameter. 

Referring  to  Eq.  (6)  of  Art.  65  : 

th 
a1  =  12  ms.,        /==  3.13  sq.  ins.,         -g-  =  0.375  sq.  ins. 

W  =  centre  weight  =  130,000  -j-  70  =  130,070  pounds. 
M  =  ^Wl  (/in  feet)  =  1,951,550. 
Hence: 

C  =     '^  '5b     =  46,400  pounds. 
42.00 

The  centre  load  did  not  break  the  beam,  but  caused  a 
deflection  of  0.9375  inch,  and  permanent  set  of  0.50  inch,  with 
beginning  of  side  deflection. 

Very  closely  approximate,  /  =  252.  Hence,  with  /  in  feet 
and  a  centre  load  of  70,000  pounds  with  the  corresponding  de- 
flection of  0.25  inch : 

E  =  36  X  70,000  x  125  =  Q        nds. 

0.25  x  252 

This  low  value  of  E  is  undoubtedly  due  to  the  fact  that 
the  beam  was  a  built  one. 


Art.  72.]  THICKNESS  OF   WEB.  6oi 

The  results  of  all  the  tests  of  built  beams  given  in  this 
chapter  show  that  they  are  much  less  stiff  than  rolled  ones  of 
the  same  section.  In  fact,  in  computing  deflections  with  the 
best  designs  and  best  quality  of  riveted  work,  E  should  prob- 
ably never  be  taken  at  more  than  about  half  its  value  for 
similar  rolled  sections,  or  say  at  12,000,000  to  15,000,000. 

After  E  is  determined,  the  deflection  at  once  results  from 
the  usual  formula  : 


/3  is  here  in  feet.      W  is  the  load  at  centre,  and  //  the  uni- 
form load  (i.  e.  weight)  of  or  on  the  beam. 


Art.  71.— Loss  of  Metal  at  Rivet  Holes. 

As  has  been  indicated  in  all  examples,  the  metal  punched 
or  drilled  from  parts  of  beams  in  tension  should  always  be 
deducted  from  the  total  tension  area  in  order  to  obtain  the 
effective  area  for  computation  of  the  ultimate  resistance.  In 
estimating  this  loss  the  actual  diameter,  as  punched  or  drilled, 
should  be  taken,  and  not  that  of  the  cold  rivet  before  driving, 
since  the  latter  is  always  at  least  one  sixteenth  inch  less  than 
the  former. 

In  the  compression  portions  of  the  beam,  if  the  work  is 
done  in  a  first-class  manner,  no  deduction  need  be  made. 


Art.  72.— Thickness  of  Web,  Plate. 

The  following  approximate  method  of  determining  the 
thickness  of  the  web  plate  in  a  flanged  beam  is  based  upon  the 
principles  established  in  Art.  28. 

It  was  shown   in  that  Article  that  on  two  planes  which 


6O2  THICKNESS  OF   Wr<3.  [Art. 


make  angles  of  90°  with  each  other  and  45°  with  the  neutral 
surface,  and  whose  intersection  forms  the  neutral  axis  at  the 
section  considered,  there  exists  on  one  a  tension  and  on  the 
other  a  compression,  each  of  whose  intensities  is  equal  to  that 
of  the  longitudinal  and  transverse  shear  at  the  same  point.  It 
was  also  shown  in  Art.  17  (see  Eq.  (38))  that  the  intensity  of 
these  shears  is  f  the  mean  intensity  of  shear  of  the  whole 
section. 

No  essential  error  is  committed  (especially  in  built  beams) 
if  it  be  assumed  that  the  whole  shear  is  taken  up  by  the  web. 
In  the  Article  just  cited  it  was  shown  that  the  intensity  of 
shear  at  the  top  and  bottom  surfaces  of  the  beam  is  zero,  as 
well  as  I  the  mean  at  the  neutral  surface.  Now,  if  this  shear 
be  assumed  uniform  in  intensity  throughout  the  transverse  sec- 
tion of  the  web,  the  shear  will  be  made  much  too  large  at  the 
top  and  bottom  surfaces,  and  only  two-thirds  its  proper  value 
at  the  centre  or  neutral  axis. 

In  accordance  with  these  assumptions  on  one  hand,  and  the 
established  principles  on  the  other,  the  web  may  be  considered 
as  composed  of  small  columns  with  ends  fixed  (at  the  flanges), 
and  with  sections  rectangular,  whose  axes  lie  at  45°  with  the 
neutral  surface. 

The  assumption  of  the  uniformity  of  shear  in  respect  to 
these  elementary  columns  causes  two  errors  in  opposite  direc- 
tions, with  the  resultant  error,  in  most  cases  at  least,  on  the 
side  of  safety. 

In  rolled  beams,  if  t'  is  the  mean  thickness  of  a  flange,  and 
*/the  total  depth,  then  the  length  of  these  elementary  columns 
may  be  taken  as  : 


or : 


/=  (d  —  2t')sec4$°. 

1=  i.4i4(</-  2t') (i) 


In  built  beams,  if  d'  is  the  depth  from  centre  to  centre  of 
rivet  holes,  there  may  be  taken : 


Art.  72.]  THICKNESS  OF   WEB.  603 


/  =    1-414^'  ........      (2) 

If  5  is  the  total  shear  at  any  transverse  section,  A  the  area 
of  that  section  of  the  web,  taking  the  depth  at  d  —  2t'  or  d't 
and  s  the  mean  shear,  or  : 


5 

S  =  -r 


then  these  elementary  columns  will  be  subjected  to  an  inten- 
sity of  compression  equal  to  s.  Hence  if  /,  the  thickness  of  a 
wrought-iron  web,  is  sufficiently  great,  there  may  be  taken,  by 
Gordon's  formula  : 


(3) 


3,000^ 
Solving  this  equation  for  / : 


=  /Y  3,000(7 -  s) (4) 

For  the  ultimate  resistance  of  wrought-iron  rectangular  col- 
umns, /  may  be  taken  at  40,000.  If  a  safety  factor  of  5  be 
taken,  the  value  of  /  becomes : 


>,ooo  —  s 


(5) 


Eq.  (5)  is  for  wrought  iron  only.  The  empirical  constants 
for  steel  yet  remain  to  be  determined. 

These  formulae  show  that  /  decreases  with  the  depth  of  the 
beam,  and  that  it  also  varies  in  the  same  direction  with  s.  If, 
therefore,  the  depth  of  the  beam  is  constant,  Eq.  (5)  need  only 


604  THICKNESS  OF    WEB.  [Art.  72. 

be  applied  at  the  section  where  s  is  the  greatest,  i.  e.,  at  or  near 
the  points  of  support. 

If,  however,  the  depth  is  variable,  it  may  be  necessary  to 
apply  the  formula  at  a  number  of  sections  in  order  to  find  the 
greatest  value  of  t. 

Eq.  (5)  frequently  gives  much  larger  values  of  t  than  are  re- 
quired. It  could  be  made  an  accurate  and  valuable  formula  if 
the  empirical  quantities  which  enter  it  were  determined  by  ex- 
periments on  flanged  beams. 

The  data  of  Art.  66  give : 

d'  =  32  ins.,         /  =  -     in.,         A  =  d't  —  14  sq.  ins. 


5  =  86,650  Ibs.,         /.         s  =  -j  =  6,200  (nearly). 

I  =  d'  x   1.414  =  45.2. 
Hence  : 

/  =  1.5  inches  (nearly). 

This  value  with  a  safety  factor  of  five  is  evidently  excessive, 
though  it  applies  only  to  the  portions  RA  and  HR  of  Fig.  I  in 
Art.  66.  Yet,  the  result  may  be  accepted  as  indicating  that 
the  web  is  certainly  too  light  for  those  portions,  and  the  ne- 
cessity of  the  stiffening  pieces  shown. 

The  data  of  Art.  67  give  : 

d'  =  27  inches,     /  =  fy&  inch,     A  —  d't  —  10.1  sq.  ins. 

S  —  26,000  pounds.     .-.     s  =  -r  =  2,600  (nearly). 

A 

I  —  d'  x   1.414  =  38.2. 


Art.  72.]  GENERAL    OBSERVATIONS.  605 

Hence  : 

/  =  0.49  inch  (nearly). 

The  thickness  taken,  therefore,  is  probably  ample,  even 
without  the  aid  of  stiffening  pieces. 

The  amount  of  assistance  to  be  derived  from  stiffeners 
cannot  be  computed  with  any  certainty.  They  are  very  essen- 
tial however,  and  should  be  introduced  in  all  large  beams. 

However  small  the  built  beam,  or  light  its  load,  the  web 
plate  should  never  be  less  than  0.25  inch  in  thickness. 

Before  leaving  this  subject  it  may  be  well  to  observe  that 
the  excessive  thickness  given  by  Eq.  (5)  was,  in  some  measure 
at  least,  to  be  anticipated.  It  has  already  been  stated  that  the 
assumption  of  uniform  compression  throughout  the  length  of 
the  elementary  column  leads  to  an  error  on  the  side  of  safety. 
Again,  the  equal  tension  at  right  angles  to  the  greatest  com- 
pression in  the  material  of  the  web,  as  well  as  the  decreasing 
compression  toward  the  centre  of  the  beam,  gives  support  to 
the  elementary  columns  throughout  their  entire  lengths. 
These  causes  give  rise  to  an  excess  of  safety,  in  the  formula, 
whose  amount  can  only  be  determined  by  experiment.  Three- 
quarters  of  the  thickness  given  by  the  formula  would  probably 
be  ample. 

The  experiments  of  the  late  Baron  vori  Weber  showed  that 
a  very  thin  web  will  give  a  remarkably  large  supporting  power. 


CHAPTER    X. 
CONNECTIONS. 

Art.  73.— Riveted  Joints. 

ALTHOUGH  riveted  joints  possess  certain  characteristics 
under  all  circumstances,  yet  those  adapted  to  boiler  and  simi- 
lar work  differ  to  some  extent  from  those  found  in  the  best 
riveted  trusses.  The  former  must  be  steam  and  water  tight, 
while  such  considerations  do  not  influence  the  design  of  the 
latter,  consequently  far  greater  pitch  may  be  found  in  riveted 
truss  work  than  in  boilers.  Again,  the  peculiar  requirements 
of  bridge  and  roof  work  frequently  demand  a  greater  overlap 
at  joints  and  different  distribution  of  rivets  than  would  be 
permissible  in  boilers. 

Kinds  of  Joints. 

Some  of  the  principal  kinds  of  joints  are  shown  in  Figs.  I 
to  6.  Fig.  I  is  a  "  lap  joint,"  single  riveted  ;  Fig.  2  is  a  "  lap 
joint,"  double  riveted  ;  Fig.  3  is  a  "  butt  joint  "  with  a  single 
butt  strap  and  single  riveted  ;  while  Figs.  4,  5  and  6  are  "  butt 
joints  with  double  butt  straps,  Fig.  4  being  single  riveted 
while  the  others  are  double  riveted.  Fig.  5  shows  zigzag 
riveting  and  Fig.  6  chain  riveting.  All  these  joints  are  de- 
signed to  resist  tension  and  to  convey  stress  from  one  single 
thickness  of  plate  to  another.  Two  or  three  other  joints 
peculiar  to  bridge  and  roof  work  will  hereafter  be  shown. 


Art.  73.] 


DISTRIBUTION  OF  STRESS. 


607 


000 


ooo 

-t- 

•>* 

ooo 

( 

p 

ooo 

c 

ooo 

c 

Fig.1 


Fig.  2 


Fig.  3 


f 

00   r^i 

f 

000 

c 

) 

)    O  O    C 

ooo 

I 

( 

) 

U    LJ 

O  0  O 

\. 

c 

ooo 

c 

) 

000 

•N           /'""•N         x~\         x- 

c 

/* 

) 

ooo 

c 

J    U    O    C 

c, 

o  o 

Fig.  4 


Fig.  5 


Fig,  6 


In  the   cases  of  bridges  and  roofs  these  "  butt  straps  "  are 
usually  called  "  cover  plates." 


Distribution  of  Stress  in  Riveted  Joints. 

A  very  little  consideration  of  the  question  will  show  that 
only  an  approximate  determination  of  the  distribution  of  stress 
in  a  riveted  joint  can  be  reached. 

In  order  that  rivets,  butt  straps  or  cover  plates,  and  differ- 
ent portions  of  the  same  main  plate  may  take  their  proper 
portions  of  stress,  an  absolutely  accurate  adjustment  of  these 
different  parts  must  be  attained  ;  but  all  shop  work  must  nee- 


608  RIVETED  JOINTS.  [Art.  73. 

essarily  be  more  or  less  imperfect,  and  the  requisite  condition 
can  never  be  maintained  during  and  after  construction.  Hence 
the  amount  of  stress  carried  by  each  rivet,  or  each  cover  plate, 
and  hence  each  portion  of  the  main  plate  at  the  joint,  cannot 
be  found. 

In  the  cases  of  lap  joints  with  three  or  more  rows  of  rivets 
(frequently  found  in  truss  work),  or  in  similar  work  when  two 
rows  of  rivets  join  a  small  plate  to  a  much  larger  one,  the  out- 
side rows,  or  row,  in  consequence  of  the  stretching  of  the  ma- 
terial at  the  joint,  must  take  far  more  than  their  portion  of 
stress,  if,  indeed,  they  do  not  carry  nearly  all.  The  same  con- 
dition of  things  will  exist  in  butt  joints  if  two  or  more  rows 
are  found,  under  similar  circumstances,  on  the  same  side  of 
the  joint. 

If  a  strip  of  plate  in  which  the  ratio  of  width  over  thickness 
is  very  considerable,  be  so  gripped  in  a  testing  machine  that 
the  applied  stress  be  approximately  uniformly  distributed  over 
its  ends,  and  if  it  be  tested  to  breaking,  it  will  be  found,  if  the 
broken  pieces  be  joined  at  the  place  of  breaking,  that  the  cen- 
tral portions  of  the  fracture  are  widely  separated,  while  the 
edges  are  in  contact.  This  is  due  to  the  cause  explained  in  Art. 
32,  "Coefficient  of  Elasticity."  Now  if  a  hole  or  holes  be 
made  in  or  near  the  centre  of  the  specimen,  a  portion  of  the 
material  in  the  front  and  rear  of  these  holes  will  be  relieved 
from  stress,  and  the  total  stress  in  the  central  section  of  the 
specimen  will  be  more  nearly  uniformly  distributed  in  the  re- 
maining material.  And  again,  these  holes  will  "neck"  the 
specimen  down  to  a  short  one.  The  influences  noticed  in  Art. 
32,  "  Ultimate  Resistance  and  Elastic  Limit"  will  thus  be  called 
into  action.  For  both  these  reasons  the  existence  of  the  hole, 
or  holes,  /;/  itself,  will  increase  the  intensity  of  the  ultimate 
resistance  of  the  plate. 

On  the  other  hand,  the  effect  of  the  punch,  if  the  hole  is 
punched,  as  will  presently  be  shown,  is  to  decrease  the  resist- 
ance of  the  metal  about  the  hole.  If  the  hole  is  in  a  joint,  also, 


Art.  73.]  BENDING  IN  JOINTS.  609 

the  bearing  pressure  between  the  rivet  and  plate  is  very  great, 
and  as  this  pressure  must  be  carried  as  tension  to  the  material 
adjacent  to  the  rivet  hole,  and  through  that  in  its  immediate 
vicinity,  the  latter  (i.e.,  the  material  at  the  extremities  of  diam- 
ters  parallel  to  the  joint)  will  receive  much  greater  tension  than 
that  in  the  central  portion  between  the  holes. 

These  last  two  influences  tend  to  reduce  the  mean  intensity 
of  ultimate  resistance  of  the  material  of  the  joint,  and  some- 
times more  than  counterbalance  the  increase  caused  by  the 
existence  of  the  holes  simply  as  such.  In  other  cases  the  re- 
sultant effect  can  only  be  determined  by  experiment. 

In  Figs.  I  and  2  it  will  be  observed  that  the  stresses  in  the 
plates  of  a  lap  joint  act  excentrically,  and,  let  it  first  be  as- 
sumed, with  a  lever  arm  equal  to  half  the  sum  of  the  thickness 
of  the  two  plates.  If,  however,  a  specimen  joint  is  put  in  a 
testing  machine,  the  resultant  stress  may  be  made  to  pass 
through  the  centre  of  the  joint,  thus  making  the  lever  arm 
for  each  plate  about  half  its  thickness. 

If,  therefore,  t  is  the  thickness  of  one  plate  and  /'  that  of 
the  other,  while  T  and  T  are  the  mean  intensities  of  tension  in 
the  plates,/  the  pitch  of  the  rivets  and  d  the  diameter  ;  in  the 
first  case  each  plate  will  be  subjected  to  the  bending  moment: 


M=  Tl(p  -  d)  =    Tt(p  -  d) 

And  in  the  second  : 

;     or,     Tt"       =-        .     .     .     (2) 


If  K  is  the  greatest  intensity  of  tensile  bending  stress,  then  : 
tM  t'M 

K=^I>    or>    17  ......    (3> 


39 


6lO  RIVETED  JOINTS.  [Art.  73. 

The  greatest  intensity  of  tension  in  the  plate  will  therefore 
be: 

T+K,     or,     T'  +  K  ......     (4) 

The  moment  of  inertia  /  will  have  the  value  : 
(P  ~  ^  (P  ~ 

~  or' 


12 

If  each  plate  has  the  same  thickness,  /  =  /'  and  T  =  T'  ; 
hence  : 

ByEq.(i)         K  =  6T  ......     (5) 

ByEq.(2)         K  =  3T  ......     (6) 

These  values  of  ^Tare  very  large  and  appear  excessive.  It 
is  to  be  remembered,  however,  that  the  formula  used  Eq.  (3)  is 
strictly  applicable  only  within  the  elastic  limit. 

There  is  no  reason  to  doubt,  therefore,  that  within  that 
limit  the  greatest  intensity  of  tension  in  the  plates  of  the  joint 
may  reach  from  4  T  to  7  T. 

From  these  considerations  it  is  to  be  expected  that  the  true 
elastic  limit  of  the  joint,  as  a  whole,  would  be  very  low. 

The  preceding  investigations  in  the  flexure  of  the  joint  are 
based  upon  the  virtual  assumption  that  the  plates  remain 
straight  after  the  application  of  external  stress.  In  reality 
such  a  condition  of  things  does  not  obtain.  Even  below  the 
elastic  limit  the  plates  begin  to  take  positions  which  are 
shown  in  an  exaggerated  manner  in  Fig.  7.  On  account  of  the 

bending,  the  material  at  the 

pj^  -JL£-  ^  _  ^  points  A  A  stretches  much 

more  than  that  at  the  points 


Fig.  7  BB  (with  low  values  of    T 

that  at  the  latter  points  may 
be  in  compression),  so  that  the  centre  lines  of  the  plates  P  and 


Art.  73-1  BENDING  IN  JOINTS.  6ll 

P'  are  brought  more  nearly  into  coincidence,  thus  lessening 
the  bending  moment  to  which  the  joint  is  subjected.  After  the 
elastic  limit  of  the  material  at  AA  is  passed,  a  considerable  in- 
crease of  strain  or  stretch  takes  place  at  those  points  for  the 
same  increment  of  stress.  Two  important  results  follow  this 
increase  of  strain  between  the  elastic  limit  and  failure  :  the 
joint  becomes  very  markedly  distorted,  so  that  the  plates  P  and 
P'  become  much  more  nearly  in  line,  and  the  stress  becomes 
much  more  nearly  uniformly  distributed  in  the  sections  AB, 
AB.  This  is  equivalent  to  saying  that  the  joint  is  subject  to  a 
greatly  decreased  bending  moment. 

If  the  plates  are  thin,  the  excess  of  strain  at  AA  over  that 
at  BB,  requisite  to  bring  the  plates  PP'  essentially  into  line, 
may  easily  be  within  the  stretching  capacity  of  the  material. 
If,  however,  the  plates  are  thick,  that  condition  will  not  hold, 
and  the  material  at  A  A  will  begin  to  fail  before  PP'  are 
nearly  in  line.  Hence,  the  mean  intensity  of  stress  in  a  thick 
plate,  other  things  being  equal  at  the  instant  of  rupture,  will 
be  considerably  less  than  that  in  a  thin  one.  It  might  thus 
happen  that  a  lap  joint  with  thin  plates  would  be  found 
stronger,  even,  than  one  with  thicker  plates. 

Reference  will  hereafter  be  made  to  experiments  which 
verify  these  conclusions. 

It  will  now  be  well  to  turn  back  a  moment  to  the  consider- 
ation of  Eqs.  (5)  and  (6).  Those  equations  show  the  effect  of 
bending  to  be  dependent  on  T  only,  and  entirely  independent  of 
the  thickness  of  the  plates,  which  apparently  contradicts  the 
conclusion  just  drawn.  But,  as  has  already  been  intimated, 
those  equations  involve  the  virtual  assumption  that  the  plates 
remain  continually  straight,  and  do  not  contemplate  the  altered 
conditions  of  the  joint  which  exist  just  at  and  before  rupture. 
Again,  they  presuppose  no  passage  of  the  elastic  limit.  There 
is  thus  no  real  contradiction. 

Although  a  single  riveted  lap  joint  only  has  been  treated, 
precisely  the  same  considerations  apply  to  a  double  riveted  lap 


6l2  RIVETED  JOINTS.  [Art.  73. 

joint,  a  butt  joint  with  single  butt  strap  or  cover  plate,  and  all 
butt  straps  or  cover  plates  of  butt  joints.  The  main  plates 
of  butt  joints  with  double  cover  plates  are  not  subjected  to 
flexure. 

The  rivets  of  all  riveted  joints  are  subjected  to  heavy  flex- 
ure, the  greatest  of  which  usually  occurs  in  single  lap  and  butt 
joints  like  Figs.  I  and  3.  An  approximate  value  of  the  bend- 
ing moment,  in  any  case,  may  be  found  as  follows : 

Let  n  be  the  number  of  rows  of  rivets  in  one  plate.  In 
Figs,  i,  2,  3,  4,  n  is  I  ;  and  2  in  Figs.  5  and  6.  Then  if  t  and  f 
are  the  thickness  of  the  two  plates  or  of  one  plate  and  one 
cover,  ?"and  T'  the  mean  intensities  of  tension  in  the  same 
pieces,  and  if  M  be  taken  from  Eq.  (i),  the  approximate  bend- 
ing moment  will  be : 

M       KAd    ,_,          .        .,  .  ,  . 

—  =   -g— ;  (From  Art.  62)  ;      ....     (7) 

in  which  A  is  the  area  of  the  cross  section  of  one  rivet,  K  the 
greatest  intensity  of  tension  or  compression  due  to  bending,  and 
d  the  rivet  diameter,  *as  before.  From  Eqs.  (7)  and  (i)  : 


lit  =  t': 

'-'   "-"\  (9) 


This  equation  is  approximate  because  it  is  virtually  assumed 
that  the  pressure  on  the  rivet  is  uniformly  distributed  along  its 
axis.*  This  is  a  considerable  deviation  from  the  truth,  particu- 

*  In  accordance  with  this  assumption,  strictly  speaking,  \t  (thickness  of  main 
plate)  should  be  taken  instead  of  t  in  the  sum  '(/  +  /')  in  the  above  formulae  for 
bending,  when  applied  to  the  double  butt  joints,  Figs.  5  and  6. 


Art.  73.]  PRESSURE   ON  RIVET.  613 

larly  as  failure  is  approached.  The  true  bending  moment  is 
much  less  than  that  given  by  Eq.  (7)  after  the  rivet  has 
deflected  a  little. 

When  the  joint  takes  the  position  shown  in  Fig.  7,  it  is  clear 
that  the  rivet  is  also  subject  to  some  direct  tension. 

There  is  a  very  high  intensity  of  pressure  between  the  shaft* 
of  the  rivet  and  the  wall  of  the  hole.  This  intensity  is  not 
uniform  over  the  surface  of  contact,  but' has  its  greatest  value 
at,  or  in  the  vicinity  of,  the  extremities  of  that  diameter  lying 
in  the  direction  of  the  stress  exerted  in  the  plate.  At  and 
near  failure  this  intensity  may  be  equal  to  the  crushing  resist- 
ance of  the  material  over  a  considerable  portion  of  the  surface 
of  contact. 

The  intricate  character  of  the  conditions  involved  renders  it 
quite  impossible  to  determine  the  law  of  the  distribution  of 
this  pressure.  The  bending  of  the  rivets  under  stress  tends  to 
a  concentration  of  the  pressure  near  the  surface  of  contact  of 
the  joined  plates,  while  the  unavoidably  varying  "fit"  of  the 
rivet  in  its  hole,  even  in  the  best  of  work,  throws  the  pressure 
towards  the  front  portion  of  the  surface  of  the  rivet  shaft.  The 
intensity  thus  varies  both  along  the  axis  and  around  the  cir- 
cumference of  the  rivet. 

If  any  arbitrary  law  is  assumed,  the  greatest  intensity  of 
pressure  is  easily  determined.  Such  laws,  however,  are  mere 
hypotheses  and  possess  no  real  value.  All  that  can  be  done  is 
to  determine,  by  experiment,  the  mean  safe  working  intensity 
on  the  diametral  plane  of  the  rivet  which  is  equivalent  to  a 
fluid  pressure  of  the  same  intensity  against  its  shaft. 

Thus,  if  f  is  this  mean  (empirically  determined)  intensity, 
d  the  diameter  of  the  rivet,  and  /  the  thickness  of  the  plate, 
the  total  pressure  carried  by  one  rivet  pressing  against  one 
plate  is: 

R=fdt (10) 

There  yet  remains  to  be  considered  the  condition  of  that 


6l4        :.--'•  RIVETED  JOINTS.  [Art.  73. 

portion  of  the  plate  on  which  the  pressure  R  =  fdt  is  applied, 
and  which  is  situated  immediately  in  front  of  the  rivet. 

This  portion  of  the  plate  is  really  in  the  condition  of  a  beam 
fixed  at  each  end,  with  a  span  equal  to  the  diameter  of  the 
rivet.  The  beam,  however,  is  not  a  straight  one.  At  each  end 
•of  the  diameter  the  direct  bending  stress  will  be  tension  ;  and, 
on  account  of  the  position  of  the  material,  its  direction  will  be 
approximately,  at  least,  that  of  the  proper  tension  of  the  plate. 
At  those  points,  therefore,  the  proper  and  bending  tension  will 
act  to  some  extent  together,  and  the  metal  will  usually  be 
more  highly  stressed  than  anywhere  else.  This  accounts  for 
the  usual  manner  of  tensile  fracture  of  a  joint,  in  which  the 
metal  begins  to  tear  on  each  side  of  the  rivets,  the  metal 
between  (generally  in  a  diagonal  direction  in  zigzag  riveting) 
being  the  last  to  give  way. 

In  the  interior  of  the  joint  it  is  quite  impossible  to  deter- 
mine the  value  of  this  tensile  bending  stress  on  each  side  of 
the  rivet.  On  the  exterior  of  the  joint,  however,  an  approxi- 
mate result  may  be  reached  ;  and  hence,  the  depth  //,  Fig.  2, 
from  the  centre  of  the  outside  row  of  rivets  to  the  edge  of  the 

plate.     The  depth  of  the  beam  will  be  taken  as  (h  -       J  ,  and 

the  pressure  or  load  will  be  considered  concentrated  at  the 
middle  of  the  diameter  or  span.  If  /  is  thickness  of  the  plate, 
/  the  pitch  of  the  rivets  and  T  the  mean  intensity  of  tension  ' 
between  the  rivets,  the  load  on  the  beam  will  be  (/  —  d}Tt\ 
and  the  moment  of  inertia  of  the  cross  section  will  be  : 


12 


From  what  has    been  shown  in    the  chapter   on    bending, 
the  modulus  of   rupture  in    the   present    case   may  be  safely 

taken  at  ^T. 

2 


*v        OF   TUI 

UNIVERSITY 


Art.  73.]  EFFECT  OF  PUNCHING. 

In  Art.  24,  the  moment  at  the  centre  and  end  of  a  span 
fixed  at  each  end  and  loaded  in  the  centre  was  shown  to  be 
equal  to  one-eighth  the  load  into  the  span. 

Hence,  by  the  usual  formulae  : 


.-.     h  =  0.71  V(p  -  d)d  +  0.5^.     .     .     .     (11) 

Reviewing  the  results  of  this  section,  it  may  be  concluded 
that  the  bending  of  the  plates  about  axes  parallel  to  them,  or 
normal  to  them  in  the  interior  of  the  joint,  and  the  bending  of 
the  rivets,  as  well  as  the  law  of  the  distribution  of  pressure 
against  them,  cannot  be  expressed  by  formula  with  any  useful 
degree  of  accuracy ;  but  that  such  influences  must  be  recog- 
nized in  the  empirical  determination  of  the  shearing  and  tear- 
ing resistances  of  the  joint  and  the  mean  intensity  of  pressure 
against  the  diametral  plane  of  the  rivet. 

Effect  of  Punching. 

The  effect  of  punching  wrought-iron  plates  has  been  found 
to  be  injurious.  The  tensile  resistance  of  the  remaining  mate- 
rial will  be  considerably  less  than  that  of  the  plate  before 
punching.  Yet  the  injurious  effect  of  the  punch  does  not  ex- 
tend far  into  the  plate.  If  the  punched  hole  is  reamed,  so  that 
the  diameter  is  increased  an  eighth  of  an  inch,  the  remaining 
plate  will  usually  give  the  normal  resistance  per  unit  of  section, 
or  essentially  so. 

It  has  been  found  by  experiment  that  effect  of  the  punch  is 
less  injurious  as  the  die  hole  is  increased  in  diameter,  although 
there  is  probably  a  limit  to  the  application  of  this  principle. 


6l6  RIVETED  JOINTS.  [Art.  73. 

The  diameter  of  the  die  hole  is  usually  from  \  to  \  larger 
than  that  of  the  punch.  This  excess  should  depend  upon  the 
thickness  (f)  of  the  plate,  and  it  is  sometimes  taken  as  o.2/. 

Numerous  foreign  experiments  (chiefly  English)  by  Barna- 
by,  Stoney,  Fletcher,  etc.,  show  that  the  loss  of  tensile  resist- 
ance due  to  punching  wrought-iron  plates  runs  usually  from 
10  to  15,  and  may  vary  from  5  to  33  per  cent,  of  the  original 
resistance. 

The  loss  of  resistance  due  to  punching  and  its  remedy,  in 
steel  plates,  have  already  been  treated  in  Art.  34. 

Wrought-iron  Lap  Joints,  and  Butt  Joints  with  Single  Butt 

Strap. 

'A  butt  joint  with  single  butt  strap,  similar  to  that  shown 
in  Fig.  3,  is  really  composed  of  two  lap  joints  in  contact ;  since 
each  half  of  the  butt  strap  or  cover  plate  with  its  underlying 
main  plate  forms  a  lap  joint.  It  is  unnecessary  therefore  to 
give  it  separate  treatment. 

From  these  considerations  it  is  clear  that  the  thickness  of 
the  butt  strap  or  cover  plate  should  be  the  same  as  that  of  the 
main  plate. 

Let  /  =  thickness  of  plates. 
"  d  =  diameter  of  rivets. 
"  p  =r  pitch  of  rivets  (/.  e.y  distance  between  centres 

in  the  same  row). 
"  T  =  mean    intensity   of   tension    in    plates  between 

rivets. 

"  T  =  mean  intensity  of  tension  in  main  plates. 
"  f  =  mean  intensity  of  pressure  on  diametral  plane 

of  rivet. 

"  5  =  mean  intensity  of  shear  in  rivets. 
"  n  —  number  of  rivets  in  one  main  plate. 
"  q  =  number  of  rows  in  one  main  plate. 
"  h  —  amount  of  extreme  lap  as  shown  in  Fig.  2. 


Art.  73.]  LAP  JOINTS.  617 

If  all  the  dimensions  are  in  inches,  then  7",  7',  /and  5  are 
in  pounds  per  square  inch. 

The  starting  point  in  the  design  of  a  joint  is  the  thickness 
/  of  the  plate.  The  rivet  diameter  is  then  expressed  in  terms 
of  /,  and  the  pitch  in  terms  of  the  diameter. 

The  thickness  /  of  boiler  plate  depends  upon  the  internal 
pressure,  and  is  to  be  determined  in  accordance  with  the  prin- 
ciples laid  down  in  Art.  9,  after  having  made  allowance  for  the 
metal  punched  out  at  the  holes  and  the  deterioration  or  other 
effect  caused  by  the  punch. 

In  truss  work  the  thickness  depends  upon  the  amount  of 
stress  to  be  carried,  and  the  same  allowances  are  to  be  made 
for  punching  and  deterioration. 

The  relation  existing  between  T  and  T'  is  shown  by  the 
following  equations: 


t(p-d)T=tfT.:        = 
or, 


In  order  that  the  joint  may  be  equally  strong  in  reference 
to  all  methods  of  failure,  the  following  series  of  equalities  must 
hold: 

-  tpT  =  -t(p-  d}  T  =  nfdt  =  o. 


.'.     tpT  =  t(p-  d)  T  =  qfdt  =  0.7854^25.     .      (13) 

It  is  probably  impossible  to  cause  these  equalities  to  exist 
in  any  actual  joint,  but  none  of  the  intensities  T\  T,for  S 
should  exceed  a  safe  working  value. 

In  ordinary  American  boiler  practice  d  varies  from   1.5^  to 


618  LAP  JOINTS.  [Art.  73. 

2t\  the  latter  for  thin  plates  and  the  former  for  thicker  ones, 
the  extreme  limits  being  about  f  inch  and  i-J-  inches. 

The  following  are  some  rules  given  by  the  best  foreign 
authorities  for  wrought  iron  : 

Browne d  =  2t  (or  1.25^  with  double  covers). . . .  (14) 

Fairbairn   d  =  it  for  plates  less  than  |  in   (15) 

Fairbairn d  —  i.^t  for  plates  greater  than  f  in ....  (16) 

Lemaitre d  =  1.5*  +  0.16    (17) 

Antoine d—  i.i\/t ..  (18) 

Pohlig d  =  2t  for  boiler  riveting '(19) 

Pohlig d  —  ^t  for  extra  strength (20) 

Redtenbacher  . .  d  —  1.5^  to  2t (21) 

Unwin d=  o.75/  +  T8T  to  \  t  +  -J (22) 

Unwin d  =  \.2^t    (23) 

As  the  results  of  some  of  his  experiments  on  ^/6-inch  steel 
plate  joints,  Prof.  A.  B.  W.  Kennedy  gives  in  "  Engineering/' 
loth  June,  1881,  the  following  rules  for  rivet  diameter : 

Single  riveted  lap  joint ....   d  =  2.2 5/  ] 

Y     -     -     (24) 
Double  riveted  lap  joint. . .   d  =  2.2\t } 

These  rules  are  for  mild  steel  plates  and  for  greatest 
strength,  but  are  not  to  be  applied  to  plates  over  }4  in.  thick ; 
as  the  diameters  would  then  become  excessive.  He  therefore 


Art.  73.] 


RIVET  DIAMETER. 


619 


DIAMETER   OF   RIVETS. 


THICKNESS 

OF 

Lloyds' 

Liverpool 

English 

French 

Wilson's 

Hovrez's 

Hall's 

PLATE. 

Rules. 

Rules. 

Dockyard 

Veritas. 

Rules. 

Rules. 

Rules. 

Rules. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

A 

£ 

fi. 

8 

1 

:  — 

f 

H 

| 

i 

f 

I 

•» 

B 

1 

ll 

i 

a. 

1$ 

£ 

— 

>i 

•|i 

1 

i 

3. 
4 

I 

i 

I 

? 

| 

¥ 

| 

I 

i 

Ifb' 

i& 

^ 

I 

|- 

It 

i 

— 

— 

i 

I 

ft 

I 

8 

i 

— 

— 

i  i} 

8 

i 

I 



i 

— 

— 

i 

I 

** 

jL 

I 

i 





^ 

I 

IH 



it 

— 

— 

1 

I 

if 

ii 

i,V 

4 

~~ 

~~ 

concluded  that  thicker  plates  than  y2  in.  would  give  propor- 
tionally less  resistance. 

It  has  been  found  by  experiment  that  there  is  a  very  de- 
cided interdependence  existing  between  the  values  of  T and/" 
in  cases  of  failure  by  tearing.  This  is  probably  due  far  more  to 
the  bending  action  of  the  rivet,  which  was  considered  in  detail 
in  one  of  the  preceding  sections,  than  to  the  direct  influence  of 
the  pressure  between  the  rivet  and  its  hole. 

Table  I.  contains  values  of  T  and/"  at  the  instant  of  failure, 
which  were  tabulated  by  Prof.  Unwin  in  "  Engineering"  for  Feb. 
20th,  1880.  All  the  plate  was  English  material.  The  results 
show  very  clearly  the  increase  of  T  with  the  decrease  of  f. 
They  are,  however,  somewhat  discordant.  The  punched  single 
riveted  lap  joints  of  Mr.  Stoney's  experiments  show  an  ap- 
parently abnormally  low  value  of  the  tenacity  T  for  a  given 
intensity  of  compression  /;  but  the  drilled  holes  show  less 
disagreement. 


62O 


LAP  JOINTS. 


[Art.  73. 


TABLE   I. 

Wrought  Iron. 


EXPERIMENTER. 

FORM  OF  JOINT. 

f,   IN   LBS.    PER 
SQUARE   INCH. 

7",   IN    LBS.    PER 
SQUARE   INCH. 

83.776 

39,650 

66,860 

44,580 

Lap  double  riveted      

78,290 

52,190 

76,830 

48.830 

F  '  b  ' 

Lap  double  riveted  

58,460 

58,460 



* 

Butt  double  riveted,  one  cover  

58,020 

53'98O 

Butt,  single  riveted,  two  covers  
Butt  single  riveted  two  covers         . 

94,210 

53^540 

Lap  single  riveted     

c8,s8o 

Butt  double  riveted  

Kirkaldy  

-^ 

Butt  double  riveted  

66 

Lap  single  riveted  

Lap  single  riveted  

8a.  080 

Browne  

• 

Butt,  single  riveted          

'  5 

SZtS^Q 

Butt  single  riveted 

'  Lap,  single  riveted,  punched  

Lap,  single  riveted,  punched  
Lap,  single  riveted,  punched  

55,660 

32,930 

Lap  single  riveted,  punched     

47,260 

Lap,  single  riveted,  punched  

41  680 

Lap,  single  riveted,  punched  

42,110 

44,350 

Stoney  ... 

40,770 

Lap  single  riveted   drilled 

64,400 

46,820 

Lap,  single  riveted,  drilled  

Lap,  single  riveted,  drilled  
Lap,  single  riveted,  drilled  
Lap  single  riveted,  drilled     

48,37° 
47,520 

36,74° 
47,490 

Lap,  single  riveted,  drilled  

Reviewing  all  the  results,  it  would  seem  that  the  following 
values  may  safely  be  given  single  riveted  lap  joints  with 
punched  holes  in  first-class  work  : 


/  =  55,000  to  60,000 
/=  5 5,000  to  50,000 


T  —  45,000  to  40,000. 
T  =  45,000  to  50,000. 


The  following  values  of/,  Tand  5,  at  the  instant  of  failure, 


Art.  73.]  PITCH  OF  RIVETS.  621 

are  from  the  experiments  (English)  of  Messrs.  Greig  and  Eyth 
and  the  Master  Mechanics'  Association. 


,/",  IN   LBS.  PER  7",  IN   LBS.    PER  S,   IN  LBS   PER 

SQ.    INCH.  SQ.    INCH.  SQ.    INCH. 

{64,4OO 46,820 40,990^ 

59,490 43,650 4I,3OO  I 

59,960 43,970 41,680  >  .  .(A) 

62,400 45,760 43,340 

66,280 47,690 38,  770  J 

All  the  holes  in  these  joints  were  drilled,  consequently,  as 
will  hereafter  be  shown,  5  is  a  little  low.  Further,  all  the  joints 
broke  by  simultaneous  shearing  of  the  rivets  a/id  tearing  of  the 
plates  :  they  may  therefore  be  considered  well  designed. 

Now,  if/=  T=  50,000,  which  is  experimentally  shown  to 
be  correct  in  single  riveted  lap  joints,  for  which  q  =  I,  the 
second  and  third  members  of  Eq.  (13)  give  : 

/  =  2d. 

But  this  pitch  would  scarcely  give  sufficient  room  for  heading 
the  rivets.  It  has  just  been  seen  that  the  results  in  group  (A) 
belong  to  well  proportioned  joints.  An  examination' of  those 
results  will  show  that  f  varies  from  1.33  T  to  1.47",  nearly; 
which  is  not  an  essential  disagreement  with  the  results  of 
Table  I.  Hence,  putting  these  values  in  Eq.  (13)  : 

/  =  2.33^    to     2.4^  ......     (25) 

This  agrees  with  good  ordinary  practice  in  boiler  making, 
which  makes  : 

/  =  2.3^    to     2.75^",  nearly. 

The  preceding  results  are  for  single  riveted  lap  joints  in  wrought 
iron. 


622 


LAP  JOINTS. 


[Art.  73. 


TABLE  II. 
Wrought  Iron  Double  Riveted  Lap   Joints. 


EXPERIMENTER   OR   AUTHORITY. 

MODE 
OF  RIVETING. 

HOLES 
MADE     BY 

POUNDS  PER  SQUARE  INCH 
FOR 

/ 

T. 

SirWm.  Fairbairn. 

Ha 
.      Mac 

nd. 
line. 

r 

? 

Pu 

D 
Pu 

nch. 

ill. 
neh. 

68,580 
70,990 
60,860 
69,490 
58,350 
5  ',030 
36,710 
56,380 
53i40o 
59^970 
57.030 
34,090 
22,020 
21,540 
21,500 

22,220 
30^330 
31,230 

5L450 
53-  '80 
45,670 
52,060 
58,350 
54,680 

57,270 
36.470 
34.670 
38,770 
45,79° 
49,060 
24,440 
23,630 
28,650 
29,610 
27,280 
28,740 

U                         I 

u             J          

u                 i               V  

David   Kirkaldy  
Easton  and  Anderson  

u             u                u 

Greig  and  Eyth  

R.  V.  J.  Knight  

U                     (I 

11            11 

11              11 

U                       It 

In  the  second  preceding  section  considerations  were  ad- 
duced which  show  that  for  a  given  value  of  the  mean  intensity 
of  compression  between  the  rivet  and  its  hole,  in  a  double 
riveted  lap  joint,  an  increased  value  of  T  over  that  for  a  single 
riveted  lap  joint  should  be  expected.  So  far  as  comparison 
can  be  made,  Tables  I.  and  II.  verify  this  conclusion,  although 

the   increase    is   not   very 

PI —          — ^ — c*~~3 ,    great-      This   arises   from 

^^r^r-  ^    the  fact  that  the  increased 

Fig.  8  length  of   a  double  joint 

requires   less   bending    at 

Ay  A,  Fig.  8,  than  a  single  one  to  bring  the  plates  P  and  P' 
nearly  into  line. 

The  tables  show  that  for  thin  plates  /  is  equal  to  7",  at  the 
instant  of  rupture,  for  an  intensity  not  far  from  55,000  pounds 


Art.  73.]  PITCH  OF  RIVETS.  623 

per   square   inch.     This  will  reduce  somewhat  the  allowable 
ratio  between /and  T. 

A  careful  examination  of  the  results  given  in  the  tables 
seems  to  make  it  perfectly  safe  to  take/  from  i.i  7"to  1.257". 
These  values  in  the  second  and  third  members  of  Eq.  (13)  give 
(remembering  that  q  is  here  equal  to  2)  for  double  riveted  lap 
joints  : 

p  =  3.2       to     3.5^) 

Or,  say:  V (26) 

/  =  3.25     to     4.0^) 

The  smaller  values  of  /  belong  to  thick  plates  and  the 
larger  values  to  thin  ones,  both  because  the  increased  thick- 
ness brings  a  greater  proportional  load  on  the  rivet  and  be- 
cause the  lever  arm  of  the  bending  moment  is  greater. 

It  should  be  stated  that  in  some  apparently  good  boiler 
practice  p  is  sometimes  -taken  as  high  even  as  $d.  The  ease 
with  which  a  double  riveted  lap  joint  is  made  steam  tight  may 
tempt  a  decrease  in  expense  of  riveting.  It  is  probable  that 
the  rivets  of  joints  in  which  the  pitch  exceeds  about  ^d  carry 
an  excessive  compression  and  a  corresponding  liability  to 
weakness. 

In  Table  II.  the  experiments  of  Mr.  Knight  were  made  on 
plates  one  inch  thick,  which  are  excessively  Heavy,  and  the  val- 
ues of /and  T  are  remarkably  small.  It  has  already  been  dem- 
onstrated that  great  thickness  of  plates  would  produce  results 
of  such  a  character,  although  the  sufficiency  of  such  an  expla- 
nation has  been  doubted.  There  seems  little  reason  to  doubt, 
however,  that  the  cause  just  cited,  together  with  the  normal 
decrease  of  resistance  with  an  increase  of  thickness,  is  a  com- 
plete explanation. 

It  is  to  be  observed  that  in  the  preceding  deduced  values  of 
f  and  7",  the  bending  of  the  plates  about  axes  both  parallel 
and  normal  to  their  surfaces,  have  been  recognized  and  pro- 
vided for. 


624  LAP  JOINTS.  [Art.  73. 

If  the  accuracy  of  the  experiments  cited  be  assumed,  and 
they  are  the  most  reliable  and  valuable  that  have  ever  been 
made,  there  may  be  taken  :. 

For       inch  plates,  T  =  30,000  to  35,000  Ibs.  per  sq.  in. 
For  i^-inch  plates,  T  =  50,000  to  55,000  Ibs.  per  sq.  in. 

And  for  intermediate  plates  proportional  values. 

For  single  riveted  lap  joints,/"  =  1.33  to  1.4    71 
For  double  riveted  lap  joints,/"  =  1. 1     to  1.25  J". 

As /"and  T  have  been  found  to  be  dependent  on  the  pecu- 
iar  circumstances  attending  the  use  of  the  material  in  the 
joint,  so,  in  the  same  general  manner,  the  determination  of  the 
ultimate  shearing  resistance  of  the  rivets  must  involve  a  similar 
recognition  of  environment. 

It  has  been  found  by  experiment,  as  might  have  been  an- 
ticipated, that  rivets  in  drilled  holes  offer  less  resistance  to 
shearing  than  those  in  punched  holes.  This  arises  from  the 
fact  that  the  edges  of  drilled  holes  are  much  sharper  than  those 
formed  by  a  punch. 

Table  III.  gives  the  mean  results  of  a  large  number  of  ex- 
periments by  the  authorities  named.  It  has  been  condensed, 
and  the  results  converted  to  pounds  per  square  inch,  from  a 
similar  one  given  by  Prof.  Unwin,  in  "  Engineering  "  for  26th 
March,  1880. 

These  results  are  for  single  riveted  lap  joints,  and  therefore 
for  single  shear.  They  are  only  a  very  little  larger  than  the 
values  determined  by  Chief  Engineer  Schock  for  single  shear, 
as  the  apparatus  of  the  latter  was  essentially  equivalent  to  a 
drilled  hole. 

For  plates  0.25  inch  to  0.375  mch  thick,  there  may  be 
taken,  as  is  usually  done,  5  =  O.8J1.  It  has  been  seen  (Table 
II.)  that  a  plate  an  inch  thick  can  be  expected,  in  lap  joints,  to 


Art.  73.] 


RIVET  SHEARING. 


625 


TABLE   III. 
Shearing  of  Wrought  Iron  Rivets. 


EXPERIMENTER   OR    AUTHORITY. 

KIND   OF   HOLE. 

S  IN    POUNDS 
PER  SQ.    IN. 

RESISTANCE  (TEN- 
SILE)  OF   PLATE 
OVER   S. 

Fairbairn    .  . 

Punched 

eo  T  80 

O    78^ 

Stoney  .        ... 

Punched. 

42  2OO 

O    OIO 

Stoney  

Drilled 

4O  Q2O 

I  061 

Fairbairn  

Punched. 

AtL    82O 

Fairbairn 

Drilled 

AT.  610 

Master  Mechanics'  Association 

Drilled 

4.6  ^QO 

Greig  &  Eyth  

Drilled. 

41  280 

I    O7I 

Mean  result                .              .  . 

Punched 

46  o^o 

o  846 

Mean  result     

Drilled 

A1!    IOO 

I  066 

give  T  not  much  over  35,000,  and  as  the  thickness  does  not 
seem  to  appreciably  affect  S,  for  this  inch  plate  there  may  be 
taken  S  =  %T.  The  ratio  of  f  over  T  has  been  seen  to  vary 
from  1.33  to  1.47".  Let  a  mean  value  of  1.36  for  this  last  ratio 
be  inserted  in  the  third  member  of  Eq.  (13) ;  then,  by  inserting 
the  other  values  just  found  in  the  fourth  member  of  the  same 
equation,  there  will  result  for  single  riveted  lap  joints  : 


For  thin  plates,  d= 
For  thick  plates,  d  = 


.       .      .      (27) 


For  double  riveted  lap  joints  these  results  would  be  dimin- 
ished only  slightly.  Hence  Eq.  (27)  may  be  taken  as  applicable 
to  both  single  and  double  riveted  lap  joints  in  wrought  iron. 

It  will  be  observed  that  Eq.  (27)  is  included  within  the  lim- 
its of  the  Eqs.  (i4)-(23). 

A  great  number  of  results  by  the  experimenters  already 
cited  in  this  chapter  show  that  the  total  resistance  of  a  single 
40 


626  LAP  JOINTS.  [Art.  73. 

riveted  lap  joint,  as  a  whole,  for  plates  not  over  0.5  inch  thick, 
may  vary  from  44  to  58  per  cent,  of  the  solid  plate  in  its  nor- 
mal condition,  and  that  the  mean  value  may  be  taken  from  50 
to  52  per  cent. 

In  a  double  riveted  lap  joint  this  mean  may  be  taken  at  60 
per  cent,  of  the  resistance  of  the  original  plate,  for  moderate 
thicknesses.  In  Mr.  Knight's  experiments  with  inch  plates 
(double  riveted),  the  resistance  of  the  joint,  as  a  whole,  ranged 
from  33  to  36  per  cent,  of  that  of  the  plate. 

It  is  clear,  from  the  preceding  investigations,  that  this 
"  efficiency "  of  the  joint  must  decrease  as  the  thickness  of 
the  plate  increases.  In  fact,  Mr.  Bertram  found,  in  1860,  that 
some  joints  in  ^-inch  plates  were  stronger  than  those  in  either 
yV  or  J^-inch  plates.  Although  such  results  do  not  involve  im- 
possibilities, they  are  certainly  remarkable,  and  have  not  since 
been  obtained. 

As  has  before  been  observed,  all  the  preceding  results  apply 
directly  to  butt  joints,  in  wrought  iron,  with  single  butt  strap  or 
cover  plate. 

The  width  of  overlap  (h)  from  the  centre  of  the  outside  line 
of  rivets  to  the  edge  of  the  plate  (see  Fig.  2)  may  now  be  deter- 
mined in  terms  of  d,  by  the  aid  of  Eq.  (n).  Since  the  load  on 
the  rivet  is  represented  by  (/  —  d}Tt,  p  must  be  taken  in 
terms  of  d  for  a  single  riveted  joint,  in  which/  =  2^Wto  2^d. 
As  a  margin  of  safety,  and  as  it  will,  at  the  same  time,  simplify 
the  resulting  expression,  let/  =  $d. 

Eq.  (11)  then  gives: 

h=  1.5^.     .......     (28) 

Experience  has  shown  that  this  rule  gives  ample  strength, 
and  is  about  right  for  caulking,  in  boiler  joints. 

The  distance  between  the  rows  of  riveting  is  not  susceptible 
of  accurate  expression  by  formulae,  although  the  considerations 
involved  in  the  establishment  of  Eq.  (11)  would  lead  to  an  ap- 


Art.  73.]  STEEL.  627 

proximate  value.  It  is  evident,  however,  that  this  distance 
should  never  be  as  small  as  //.  Apparently,  in  more  than 
double  riveted  joints,  this  distance  should  increase  as  the  centre 
line  of  the  joint  is  receded  from,  in  consequence  of  the  bending 
action  of  .the  rivet.  There  are  other  reasons,  however,  besides 
that  of  inconvenience,  why  such  a  practice  is  not  advisable. 

In  chain  riveting  the  distance  between  the  centre  lines  of  the 
rows  of  rivets  may  be  taken  equal  to  the  pitch  in  a  single  riveted 
joint,  or,  as  a  mean,  at  2.5  the  diameter  of  a  rivet. 

In  zigzag  riveting  (Fig.  5)  this  distance  may  be  taken  at 
three-quarters  its  value  for  chain  riveting. 


Steel  Lap  Joints  and  Butt  Joints  with  One  Cover. 

The  general  phenomena  attending  the  tests  of  steel  joints 
are  precisely  the  same  in  kind  with  those  observed  in  connec- 
tion with  riveted  iron  plates  ;  they  do  not,  therefore,  need  par- 
ticular consideration  in  this  section. 

Table  IV.  contains  results  communicated  to  the  "  Commit- 
tee of  the  Institution  of  Mechanical  Engineers "  by  Messrs. 
Parker  and  Sharp  ("  Engineering,"  i6th  April,  1880).  The 
joints  failed  by  tearing,  and  gave  the  values  of  T  shown  in  the 
table.  The  intensity  of  pressure,/*,  existed  at  rupture. 

The  following  values  of  T  and/  under  precisely  the  same 
circumstances,  i.e.,  failure,  were  found  by  Prof.  A.  B.  W.  Ken- 
nedy ("  Engineering,"  2Oth  May  and  loth  June,  1881,)  for  single 
riveted  lap  joints. 


THICKNESS 

OF   I' LATE.                                     T.  f. 

8-inch 67,060  Ibs.  per  sq.  in 42,980  Ibs.  per  sq.  in. 

1     65,310   "  "  "  " 57,6oo  "  "  "  " 

i     "     77,050"  "  "  " 70,850"  "  "  " 

i     "    73,030    "  "  "  " 70,520  "  "  "  " 

f    "     80,920"  "  "  " 73,420"  "  "  " 


628 


STEEL  LAP  JOINTS. 


[Art.  73, 


TABLE  IV. 
Steel  Joints. 


JOINT. 

HOLE. 

THICKNESS 
OF   PLATE. 

POUNDS   PER  SQ.   IN.    FOR 

T. 

/. 

Tre 

Qus 
Doi 
Doi 

ble  rive 

tdruple 
ible  riv 
ible  riv< 

ted  (ch 

riveted 
2ted  bu 
ited  bu 

a  in)  .  . 

Drilled. 
Punched. 

Drilled. 

p 
7 
Drilled. 
Punched. 

1   in. 
1  in. 
M,  jn. 
i  in. 
i  in. 
i  in. 
4   in. 
i  in. 
4-  in. 
I   in. 
|  in. 
4  in. 
I   in. 
*  in. 

? 

79,220 

52,280 
50,330 
73,360 

70,040 
80,890 
78,400 
66,940 

67,520 
80,380 
75,780 
68,250 
65,950 
56,760 

87,920 

97,730 

60,010 

39-380 
37,740 
57,700 
55,080 

54,470 
52,790 
52,220 
53,330 
73,830 
49,500 
47,58o 
35,46o 
42,360 
76,200 
83,840 

(zigzajr)  «  • 

tt  (one  cover)  .  .  . 
tt  (one  cover)  .  .  . 

The  holes  in  these  plates  were  all  drilled,  and  each  result  is 
a  mean  of  two  tests. 

These  experiments  do  not  present  a  sufficient  range  to 
show  clearly  the  relation  existing  at  failure  between  T  and/. 
It  is  clear,  however,  that  no  recorded  intensity /has  been  large 
enough  to  decrease  T  to  any  appreciable  amount.  In  some  of 
Prof.  Kennedy's  tests,  in  which  failure  took  place  by  shearing, 
/was  not  far  from  1.27"  (with  T  —  65,000  to  75,000),  and  it 
would  appear  from  his  experiments  that  such  a  ratio  may  prop- 
erly be  taken  for  thin  plates  in  single  riveted  joints.  At  the 
same  time,  with  the  mild  steel  used  by  Prof.  Kennedy,  T  may 
be  taken  at  70,000  pounds  for  plates  Y±  to  y%  inch  thick. 

Putting  i.2jTfor/  in  the  third  member  of  Eq.  (13): 


Art.  73.]  PITCH  OF  RIVETS.  629 

/  =  2.2d     } 

Or,  say,  \ (29) 

/=  2.25^) 

for  single  riveted  lap  joints.  It  .will  probably  be  best  to  allow 
this  pitch  to  stand  for  thick  plates  also,  although  experiments 
to  verify  such  a  conclusion  are  yet  lacking.  For  very  thick 
plates  in  single  riveting,  however,  T  should  not  be  taken  over 
50,000  to  55,000  pounds  at  the  highest. 

Experiments  on  double  riveted  lap  joints  by  Martell,  Kir- 
kaldy  and  Easton  and  Anderson,  show  that  it  will  be  essentially 
correct,  and  certainly  safe,  to  take  f  and  T  as  in  the  single 
riveted  joints.  With  q  equal  to  2,  Eq.  (13)  will  then  give  for 
double  riveted  steel  lap  joints  : 


Or,  say,  \ (30) 


Although  relating  to  treble  and  quadruple  riveted  joints, 
Table  IV.  shows  in  a  marked  manner  the  decrease  of  T  with 
the  increase  of  thickness,  and  verifies  the  conclusion  drawn  in 
the  preceding  section  in  regard  to  that  phenomenon. 

The  results  cited  by  Prof.  Unwin,  in  the  report  so  fre- 
quently referred  to  heretofore,  indicate  that  for  treble  riveted 
joints  /  may  be  taken  essentially  equal  to  T  for  thin  plates, 
and  0.92"  for  thiek  ones.  Hence,  using  Eq.  (13)  as  before : 

TREBLE    RIVETING. 

Thin  plates  (0.25  and  0.375  in.),    p  —  ^d     ) 

'      [    -    -    (30 
Thick  plates  (0.875  and  i.oo  in.),/  =  3.7^) 

Some  experiments  of  Mr.  Kirkaldy  on  joints  with  J^-inch 
Siemens  steel  plates  quadruple  riveted,  seem  to  show  that  the 
pitch  should  be  about  the  same  as  in  treble  riveted.  This  is 


630 


STEEL  LAP  JOINTS. 


[Art.  73. 


undoubtedly  due  to  the  fact  that  with  such  a  great  number  of 
rivets  it  becomes  impossible  to  obtain  even  an  approximately 
proper  distribution  of  load  among  them. 

In  treble  and  quadruple  riveting  the  tests  cited  show  that 
Tmay  be  taken  at  70,000  to  75,000  for  thin  plates,  and  55,000 
to  60,000  for  thick  ones. 

In  all  the  preceding  investigations  it  is  supposed  that  the 
holes  are  drilled,  or  that  the  plates  are  subsequently  annealed 
if  punched. 

In  nearly  all  the  experiments  cited  by  Prof.  Unwin,  the 
value  of  T,  as  found  in  the  actual  joint,  exceeded  the  ultimate 
resistance  of  the  original  plate  ;  a  result  which  finds  its  ex- 
planation in  the  drilling  of  the  holes  and  the  "  shortening  " 
effect  produced  by  their  presence,  aided  by  their  equalizing 
effect. 

Table  V.  gives  the  ultimate  shearing  resistance  of  steel 
rivets  as  determined  by  Sharp,  Martell,  Kirkaldy  and  Greig 
and  Eyth.  A  very  considerable  reduction  is  noticed  with  the 
increase  in  plate  thickness,  due  probably  to  increased  bending 
and  size  of  rivet. 

Prof.  Kennedy  found  the  following  values  in  single  riveted 
lap  joints : 


RIVET    DIAM. 


s. 


0.75  in 54,460  Ibs.  per  sq.  in. 

i.oo  " 37>24° 

i  .00  " 38,720 

0.75  " 48,030 

o-75  " 49»450 

o. 75  " 49>48o 

0-75  " 49.300 

0-75  "t 47,870 


Each  result  is  a  mean  of  two  or  three  tests. 

In  Mr.  Kirkaldy's  four  tests  of  ^6-inch  treble  and  quadruple 
riveted  lap  joints,  with  i^-inch  rivets,  the  ultimate  shearing 
resistance  S  varied  from  41,110  to  46,260  Ibs.  per  sq.  in. 


Art.  73  J 


RIVET  SHEARING. 


TABLE  V. 
Shearing  of  Steel  Rivets. 

JOINT. 

MEAN   OF. 

THICKNESS 
OF    PLATE. 

S  IN   POUNDS   PER 
SQ.    IN. 

Single  rive 
Double  rive 
Treble 

Quadruple 

ed  

2 

6 

8 

i 
7 

i*u 

! 

i 

57-570 
53»690 

53,310 
50,650 
60,930 
56,220 

57>i2o 
53,540 
53,980 
43,56o 
46,140 
43,010 

'ted  (chain)  

riveted  (zisrzacr) 

Four  experiments  by  Mr.  Kirkaldy  on   single  riveted  lap  joints,  during  1881, 
gave  S  varying  from  52,106  to  54,042  Ibs.  per  sq.  in. 

Prof.  Kennedy's  results  give  nearly: 

S  =  o.?T. 
Tables  IV.  and  V.,  plates  not  over  ^  in.  thick : 

5  =  0.8  r. 
Mr.  Kirkaldy's  for  treble  and  quadruple  riveting : 

5=  0.7  T. 

For  ordinary  plates  therefore  in  single  and  double  riveting, 
for  which  /  =  i.27~and  5  as  a  mean  —  0.75  7",  the  third  and 
fourth  members  of  Eq.  (13)  give  : 

d  =  2t  (nearly) (32) 


632  BUTT  JOINTS.  [Art.  73. 

For  thick  plates  in  treble  and  quadruple  riveting,  for  which 
/=  0.97",  and  5  =  Q.*jT\ 

d  —  i. 6t  (nearly) (33) 

The  rivet  pitch,  therefore,  for  steel  plates,  may  be  said  to 
vary  from  2t  for  thin  plates  to  i.6t  for  thick  ones,  with  a 
maximum  diameter  of  ij  to  i^  inches. 

Prof.  Kennedy's  best  designed  single  riveted  lap  joints 
gave  from  55  to  64  per  cent,  the  strength  of  the  solid  plates. 

Well  designed  double  riveted  lap  joints  should  give  from 
65  to  75  per  cent,  the  resistance  of  the  solid  plate. 

Equally  well  constructed  treble  and  quadruple  riveted 
joints  should  have  an  efficiency  of  70  to  80  per  cent,  of  the 
solid  plate. 

It  is  therefore  seen  that  there  is  little  economy  in  more 
than  double  riveting  ordinary  joints. 

The  distance  between  the  centre  lines  of  the  rows  of  rivets, 
and  the  distance  from  the  edge  of  the  lap  to  the  outside  centre 
line  of  holes,  may  be  taken  the  same  as  for  wrought-iron 
joints,  according  to  the  rules  given  in  the  last  part  of  the  pre- 
ceding section. 

All  rivets  have  heretofore  been  supposed  to  be  steel.  In 
the  case  of  steel  plates  and  iron  rivets,  there  may  be  taken,  at 
least  approximately,  0.95  for  5,  and  f  =  T  for  thin  plates,  or 
O.87"  for  very  thick  ones.  These  values  are  to  be  inserted  in 
the  preceding  formulae  for  all  steel  joints,  and  the  results  for 
/  and  d  taken. 

Wrought-iron  Butt  Joints  with  Double  Covers. 

Butt  joints  with  double  butt  straps  or  covers  differ  in  two 
respects,  and  advantageously,  from  lap  joints  and  butt  joints 
with  a  single  cover ;  2.  e.,  in  the  former  the  rivets  are  in  double 
shear  and  the  main  plates  are  subjected  to  no  bending.  The 


Art.  73.]  THICKNESS  AND  PITCH.  633 

cover  plates,  however,  are  subjected  to  greater  flexure  than  the 
plates  of  a  lap  joint,  for  there  is  no  opportunity  to  decrease 
the  leverage  by  stretching.  As  the  covers  form  only  a  small 
portion  of  the  total  material,  these,  with  economy,  may  be 
made  sufficiently  thick  to  resist  this  tendency  to  failure. 

Let  t'  —  thickness  of  each  cover  plate. 

And  let  the  remaining  notation  be  the  same  as  in  the  pre- 
ceding section.  The  intensity  of  compression  between  the 
walls  of  the  holes  in  the  cover  plates  and  the  rivets,  and  the 
tension  in  the  former,  will  be  ignored  on  account  of  the  excess 
in  thickness  of  the  two  cover  plates  combined  over  that  of  the 
main  plate.  This  excess  in  thickness  is  required  on  account  of 
the  bending  in  the  covers  noticed  above. 

The  thickness  of  each  cover  should  be  front  ^  to  Tfe  the  tJiick- 
ness  of  the  main  plates,  or  /'  =  0.75^  to  0.875^. 

The  combined  thickness  of  the  covers  will  thus  be  from 
1.50  to  1.75  that  of  the  main  plates. 

The  four  principal  methods  of  rupture  in  the  main  plate 
will  then  lead  to  the  following  equations,  corresponding  to 
Eq.  (I3): 

-  tpT  =  -t(p-d)  T=  nfdt  =  1.5708^*5. 

.-.  tpT  =  t  (p  -  d)  T  =  qfdt  =  1.5708^5     .     (34) 

The  experiments  of  Kirkaldy,  Fairbairn,  Greig  and  Eyth 
and  Knight,  show  that  in  well  proportioned  joints  f  =  1.25  to 
1. 5  T  (the  higher  values  belonging  to  the  thinner  plates),  with 
a  mean  value  of  about  1.47".  As  no  bending  exists  in  the 
main  plates,  this  value  holds  in  single  or  double  riveting. 

Hence  for  single  riveting,  the  second  and  third  members  of 
Eq.  (34)  give 

p  =  2.4^;     or,  say,    /  =  2.5^   ....     (35) 


634  BUTT  JOINTS.  [Art.  73. 

In  double  riveting,  for  which  q  —  2  : 

/  =  3.8^;     or,  say,    /  =  4.0^    ....     (36) 

On  account  of  the  essential  impossibility  of  even  an  ap- 
proximately proper  distribution  of  the  load  among  the  rivets, 
and  the  consequent  liability  of  failure  of  the  joint  in  detail,  in 
treble  riveting  the  pitch  should  probably  not  exceed  4.5^,  nor 
%d  in  quadruple  riveting. 

There  may  be  taken,  according  to  the  experiments  just 
cited  : 

For  punched  inch  plates  : 

T  =  40,000  Ibs.  per  square  inch. 
For  drilled  J^-inch  plates  : 

T  =  55,000  Ibs.  per  square  inch. 

Other  thicknesses  and  conditions  give  approximately  pro- 
portional values,  allowing  about  10  per  cent,  for  the  deteriora- 
tion of  the  punch  ;  i.e.t  T,  for  a  $/$  punched  plate,  may  be  taken 
at  45,000  pounds. 

It  has  already  been  observed  that  the  value  of  vS  may  be 
taken  at  0.8  T  for  lap  joints,  but  the  few  experiments  that  have 
been  made  on  shearing  in  butt  joints  with  double  covers,  show 
that  the  ratio  must  be  taken  somewhat  less,  in  consequence 
probably  of  the  double  shearing  which  takes  place. 

Hence,  let  5  be  taken  at  0.75  T. 

Using  the  third  and  fourth  members  of  Eq.  (34),  therefore, 
and  making  5  =  0.75  T  : 

For  thin  plates  in  which  f  —  1.5  Z ' : 

<*=  1-3' (37) 


Art.  73.]  STEEL  BUTT  JOINTS.  635 

For  thick  plates  in  which  f  =  1.257": 

<*=  LI*  ........     (38) 

It  is  hardly  worth  while,  however,  to   make  any  rivet  less 
than   y%  inch   in   diameter.     Hence  there    may  be  taken  the 

limits  : 


For  i^-inch  plate  ;  d  =  0.375  inch. 
For    i-inch  plate  ;  d  =  1.125  inch. 

These  results  are  verified  by  good  boiler  practice. 

The  distance  from  the  centre  line  of  outside  row  of  rivets 
to  the  edge  of  the  cover  plate,  or  from  the  edge  of  the  main 
plate  to  the  centre  line  of  the  first  row  of  rivets  in  the  same, 
may  be  taken  at  \d  as  in  lap  joints,  since  the  calculation  is 
precisely  the  same.  This  rule  frequently  gives  a  considerable 
margin  of  safety  over  that  of  any  other  portion  of  the  joint. 

The  distance  between  the  centre  lines  of  the  rows  of  rivets 
may  be  taken  at  2.5  to  3.0^  for  chain  riveting,  and  ^  that  dis- 
tance for  zigzag  riveting. 


Steel  Butt  Joints   with  Double  Cover  Plates. 

For  the  same  reasons  stated  in  the  preceding  section,  con- 
siderations touching  the  stress  in  the  cover  plates  will  be 
omitted.  And  also,  for  the  reasons  there  given,  these  cover 
plates  should  each  possess  from  ^  to  ^  the  thickness  of  the 
main  plate  ;  or  : 

t'  =  0.75     to    0.875/. 

Table  VI.  gives  the  results  of  a  large  number  of  tests  in 
which  the  joint  failed  by  the  tearing  of  the  plates.  The  in- 
tensities of  tension  and  compression,  T  and/,  existed  at  failure. 


636 


STEEL  BUTT  JOINTS. 


[Art.  73. 


TABLE   VI. 

Double  Riveted  Butt  Joints. 


KXPERIMENTER  OR   AUTHORITY. 

HOLES   BY 

POUNDS   PER  SQUARE   INCH 
FOR 

T. 

/. 

Henry  Sharp                               .  . 

Drill. 
Punch. 
Drill. 

ii 

Punch. 

a 

ii 

M 

Drill. 

96,160 
87,600 
55,ioo 
5i,740 
64,290 
58,690 
55-200 
§1,230 
4,320 
68,990 

75,49° 
82,450 
83,180 

76,59° 
78,220 
74,030 
70,540 
73,920 
72,560 
72,39° 
76,520 
67,670 
57,36o 
49070 
50,920 
62,140 
61,800 
66,200 
63,260 
63,560 
69,590 
67,540 
66,750 
67,260 

83,330 
75,  i  7° 
76,205 
71,680 
88,890 
89,130 
76,160 
70,800 
88,930 
93,160 
101,900 
99,660 
100,510 
90,460 
92,380 
92,850 

84,630 
83,080 
107,110 
112,780 
92,270 

7i,440 
49,100 
50,760 
107,410 
74,520 

112,000 
107,700 
104,100 
117,300 
98,630 
108,000 
I2I,o6o 

Martell  ....                        .               .                    .     . 

Boyd            .     .               .     .     

Kirkaldy,  annealed  plates      

Greig  and  Eyth  .  .  . 

Parker  

"                                                          

Kirkaldy,  \  inch  Siemens  steel  plates.    Mean  of  two. 
A    "     Landore     "                If  rivets  

"           "        "    "  ::::  : 

Bored. 
Punch. 
Bored. 

The  first  of  the  last  set  of  results  in  the  table,  by  Mr.  Kirkaldy, 
was  found  with  zigzag  riveting  in  which  the  distance  between 
the  centre  lines  of  the  rows  of  rivets  was  too  small. 

These  results  are  quite  irregular,  but  it  would  seem  to  be  as 
safe  a  deduction  as  possible  to  take/  =  1.257;  with  T  equal  to 
70,000  to  75,000  pounds  per  square  inch  for  thin  plates,  and 
55,ooo  to  60,000  for  thick  ones. 


Art.  73.]  PITCH  AND  RIVET  SHEARING.  637 

With  this  value  of/,  and  q  —  2,  the  second  and  third  mem- 
bers of  Eq.  (34)  give  for  double  riveted  butt  joints  with  two 
covers  : 

P  =  3-5^.     .......     (39) 

If  the  same  value  of  f  be  preserved,  there  will  result  for 
single  riveted  butt  joints  with  two  covers  : 

P  =  2.5<*     ......     (40) 

Experiments  on  treble  and  quadruple  riveting  are  yet  lack- 
ing. 

But  few  experiments  on  the  shearing  of  rivets  in  butt 
joints  with  double  covers  have  yet  been  made.  Four  tests  by 
Messrs.  Sharp  and  Kirkaldy  give  : 

THICKNESS 

OF   PLATE.  S. 

Single  riveted  ..........    -       ..........  42,000  Ibs.  per  sq.  in. 

Double  "  ..........  0.875  in  ..........  44.55°  "  "  "  " 

..........    -       ..........  53,370    "      "     "    " 

"  "  ..........  0.55  in  ..........  42,700  "  "  "  " 

..........  o.875in  ..........  44,420    "      "    "    " 

All  the  holes  were  drilled. 

These  values  of  5  range  about  0.77".  Putting  this  ratio, 
therefore,  in  Eq.  (34),  and  taking  /=  1.257;  the  third  and 
fourth  members  of  that  equation  give  : 


(40 


It  is  probable  that  this  is  a  little  too  small  for  thin  plates, 
and  a  little  too  large  for  thick  ones.  Hence  there  may  be 
taken  : 

For  thin  plates,  d  =  \%t\ 

\      .....     (42) 
For  thick  plates,  d  =  i%t) 


638 


EFFICIENCIES  OF  JOINTS. 


[Art.  73. 


Double  riveted  butt  joints  designed  in  accordance  with  the 
foregoing  deductions  should  give  a  resistance  ranging  from  65 
to  75  per  cent,  of  that  of  the  solid  plate. 

Single  riveted  joints  will  give  an  efficiency  somewhat  less ; 
perhaps  from  60  to  65  per  cent. 

It  is  to  be  supposed,  in  applying  the  rules  just  established, 
that  all  steel  plates  are  drilled,  or  subsequently  annealed  if 
punched. 

As  in  the  preceding  cases,  the  distance  between  the  centre 
lines  of  the  rows  of  rivets  may  be  taken  at  2.5  to  ^d  for  chain 
riveting,  and  three-quarters  that  distance  for  zigzag. 


Efficiencies. 

The  values  of  the  quantity  which  has  been  termed  the 
"efficiency  "of  the  joint,  i.e.,  the  ratio  of  the  resistance  of  a 
given  width  of  joint  over  that  of  an  equal  width  of  solid  plate, 
in  the  preceding  investigations,  are  those  actually  determined 
by  experiments  with  the  joints  themselves.  They  may,  there- 
fore, be  relied  upon.  Some  values  which  have  for  many  years 
been  considered  as  standard,  but  which,  in  reality,  are  of  a 

TABLE   VII. 
Butt  Joints  with  Two  Covers — 1877. 


NO.  OF 

TESTS. 

PLATE 
THICKNESS. 

RIVET  DI- 
AMETER. 

PITCH   OF 
RIVETS. 

HOLES. 

RIVETING. 

EFFICIENCY. 

2 

•]TS  in. 

fin. 

2J-  in. 

Punched. 

Chain. 

0.672 

2 

i\-  in. 

&in. 

3    in. 

Punched. 

Zigzag. 

0.669 

2 

i  in. 

fin. 

24  in. 

Drilled. 

Chain. 

0.662 

2 

i  in. 

lin. 

3    in. 

Drilled. 

Zigzag. 

0.633 

Art.  73.] 


EFFICIENCIES  OF  JOINTS. 


639 


somewhat  arbitrary  nature,  and  at  best  belonging  to  a  limited 
class  of  joints,  have  been  disregarded. 

Table  VII.  gives  the  results  of  Mr.  Kirkaldy's  experiments 
in  reference  to  the  comparative  resistance  of  chain  and  zigzag 
riveting.  The  difference  is  not  great,  but  what  there  is  is  in 
favor  of  the  chain  riveting. 

TABLE   VIII. 

Kirkaldy's  Tests— 1872. 


JOINT. 

RIVETING. 

HOLES. 

RIVET    DIAMETER 
IN  TERMS  OF  t. 

PITCH   IN 
TERMS   OF  d. 

EFFICIENCY. 

Lao 

Single 

Punched 

d  —  2t 

~     d 

O    ZZ 

Single 

Drilled 

d  —  2t 

•b  —  2$d 

v-  Da 
o  62 

Lap  .  . 

Double 

Punched 

d  —  2t 

o  69 

T        V    

JLap 

Double 

Drilled 

d  —  2t 

*  ^ 

O    7£ 

Butt    I  cover  

Single 

Punched 

d  —  2t 

6  —    d 

u.  /u 

Oc  e 

Butt,  I  cover  

Single 

Drilled. 

d  —  2t     ' 

•b  —  2*.d 

o  62 

Butt    i  cover 

Double 

d  —  2t 

?     -  *f,u 

Butt    i  cover    

Double 

Drilled 

d  —  2t 

S>—// 

O*»r 

Butt,  2  covers  
Butt    2  covers 

Single. 

Punched. 
Drilled 

d  —  i\t 
d  —  I1/ 

P  =  *\d 

0-57 

o  67 

Butt    2  covers  .    ... 

Double 

Punched 

d  —  i'/ 

*  Z  Jij 

O72 

Butt,  2  covers  

Double. 

Drilled. 

v=1f/ 

f=*v 

0.79 

Table  VIII.  gives  the  results  of  the  same  experimenter  on 
the  relative  value  of  punched  and  drilled  work. 

The  drilled  work  is  seen  to  give  decidedly  the  greatest 
efficiency  in  every  case. 

The  joints  to  which  Tables  VII.  and  VIII.  belong  were  of 
wrought  iron. 

Experiments  by  Mr.  Kirkaldy  during  1881  show  that  well- 
designed  double  riveted  steel  butt  joints  with  two  covers  may 
be  expected  to  give  efficiencies  varying  from  0.65  to  0.75. 


640 


TRUSS  JOINTS. 


[Art.  73, 


Riveted  Truss  Joints. 

The  circumstances  in  which  riveted  joints  are  used  in  truss 
work,  render  permissible  many  special  forms  which  can  find  no 
place  in  boiler  riveting.  If  joints  are  found  under  the  same 
circumstances,  so  far  as  the  transference  of  stress  is  concerned, 
precisely  the  same  forms  would  be  used,  except  that  caulking 
is,  of  course,  only  required  in  boiler  work. 

Fig.  9  shows  a  common  form  of  chord  construction  in  riv- 


r\    r\  r\ 


1                                                                                   1     r\ 

^-^ 

\                           r'                        D 

5 

N                                 0'                            ,D 

/ 

i 

A 


Fig.9 

eted  truss  work,  with  the  relative  proportions  A 
exaggerated. 

The  lower  portion  of  the  figure  shows  a 
section  of  the  chord,  in  which  the  cover 
plates  are  shaded.  The  joint  is  supposed  to 
be  in  tension. 

AB  is  a  horizontal  cover  plate,  under 
which  the  horizontal  component  plates  form 
lap  joints  at  C,  D  and  E.  As  the  distance 
MN  must  necessarily  be  much  greater  than  the  allowable  pitch 
in  boiler  work,  these  lap  joints,  considered  in  themselves, 
should  be  at  least  treble  riveted.  On  the  other  hand,  the  pre- 
ceding investigations  show  that  even  with  treble  riveting  there 
is  great  disparity  in  the  loads  carried  by  the  different  rivets 
and  consequent  tendency  to  detailed  rupture  ;  there  would 


^ 

j 

Y 

C 

) 

£ 

) 

q 

) 

F     G 


Art.  73.]  TRUSS  JOINTS.  641 

seem,  therefore,  to  be  little  or  no  benefit  in  more  than  treble 
riveting. 

The  distance  between  the  centres  of  rivets  along  the  line  of 
the  chord — i.e.,  along  AB  in  the  upper  figure — may  be  taken  at 
three  diameters.  The  overlap  CD  —  DE  (upper  Fig.)  would 
then  be  taken  at  9  diameters,  and  from  A,  C,  D  or  E  to  the 
centre  of  the  first  hole,  at  \y2  diameters.  The  cover  AB 
should  extend  9  diameters  also  on  either  side  of  C  and  E. 

In  this  work  the  diameter  of  the  rivet  may  usually  be  taken 
about  the  same  as  for  boiler  work.  In  .estimating  the  resist- 
ance of  the  whole  joint,  however,  it  is  to  be  borne  in  mind 
that  the  rivet  holes  take  metal  out  of  all  the  plates,  and  that 
they  are  usually  punched. 

It  is  impossible  to  follow  the  stresses  in  such  a  joint  or  to 
compute  its  efficiency.  If  tested  to  failure,  the  latter  would 
probably  be  found  pretty  low. 

The  joint  in  the  vertical  plate  should  be  formed  as  at  FG — 
i.e.,  it  should  be  a  double  cover  butt  joint.  The  principles  al- 
ready established  in  a  preceding  section,  in  regard  to  the  thick- 
ness of  covers  and  diameter  of  rivets,  should  be  observed  here. 

The  two  rows  of  rivets  on  either  side  of  the  joint  may  as 
well  be  chain  riveted  with  a  pitch  3^  to  4  diameters.  Other 
rivets  should  then  be  staggered  in  until  the  group  of  rivet  cen- 
tres on  each  side  is  brought  to  a  point,  as  shown  in  the  up- 
part  of  Fig.  9.  In  this  manner  the  available  section  of  a  width 
of  plate  equal  to  that  of  the  cover,  becomes  approximately 
equal  to  the  total,  less  the  material  from  one  rivet  hole.  Hence 
the  efficiency  of  the  joint  becomes  correspondingly  increased. 

If  the  joint  is  in  compression  the  preceding  observations 
hold  without  change,  except  that  all  covers  should  have  the 
same  thickness  as  the  plates  covered. 

Even  if  the  joints  C,  D,  E  and  H  are  of  planed  edges,  little  or 

no  reliance  should  be  placed  upon  their  bearing  on  each  other, 

since  the  operation  of  riveting  will  draw  them  apart   more  or 

less,  however  well  the  work  may  be  done.     Melted  zinc,  or 

41 


642  FRICTION  OF  RIVETED  JOINTS.  [Art.  73. 

other  similar  metal,  has  been  poured  into  compression  joints 
with  the  intention  of  insuring  good  bearings,  but  the  results 
are  not  satisfactory* 

In  the  case  of  very  wide  chords,  four  longitudinal  rows  of 
rivets  should  be  used  in  such  joints  as  are  exemplified  in 
Fig.  9. 

Unless  great  caution  is  observed  and  excellence  of  design 
secured,  there  will  frequently  be  excessive  bending  in  the 
riveted  joints  of  trusswork,  on  account  of  the  great  variety  of 
connections  required. 

Diagonal  Joints. 

It  has  been  proposed  to  form  riveted  joints,  the  edges  of 
whose  plates  are  neither  perpendicular  nor  parallel  to  the 
stress  transferred.  In  this  manner  a  greater  number  of  rivets 
and  a  greater  section  of  metal  will  resist  the  stress  exerted  in 
the  body  of  the  plate. 

Mr.  Kirkaldy  made  some  tests  on  such  lap  joints,  single 
riveted,  with  ^-inch  plates,  the  joints  of  which  lay  at  45°  with 
the  applied  force,  with  the  following  results  : 

Entire  plate 100 

Square  joint 59-4 

Diagonal  joint 87.2 

The  diagonal  joints  are  thus  seen  to  give  by  far  the  best 
results.  They  are,  however,  much  the  most  expensive  also. 


Friction  of  Riveted  Joints. 

There  are  not  lacking  experiments  to  show  that  the  friction 
between  the  plates  of  a  riveted  joint  is  very  great.  This,  how- 
ever, cannot  be  relied  upon  to  give  additional  resistance  to  the 
joint,  since  a  sensible  relative  movement  of  the  plates  takes 


Art.  74.]  WELDED  JOINTS.  643 

place  in  advance  of  its  greatest  resistance  and  essentially  de- 
stroys the  friction. 

The  experiments  of  Edwin  Clarke,  Harkort  and  Lavelley 
show  that  this  friction  may  range  from  8,330  to  22,400  Ibs. 
per  sq.  in.  of  rivet  section. 

The  specimens  were  prepared  with  one  slotted  plate,  so 
that  friction  was  the  only  resistance  to  the  parting  of  the 
plates. 

• 
Hand  and  Machine  Riveting. 

Pneumatic,  steam  and  hydraulic  riveting  machines  have 
lately  been  brought  to  such  a  degree  of  perfection,  that  ma- 
chine work  is  now  very  generally  preferred  to  hand  riveting. 

The  resistances  of  joints  will  vary  to  some  extent  with  the 
method  of  riveting.  Usually,  however,  the  variation  will  not 
be  greater  than  may  be  found  for  the  same  kind  of  riveting  in 
different  places  and  under  different  circumstances. 

As  a  rule,  machine  riveting  is  much  more  reliable  than 
hand,  in  that  the  hole  is  better  filled  and  the  rivet  more 
quickly  headed,  in  consequence  of  the  great  excess  of  pressure 
exerted.  There  is  thus  much  less  liability  of  loose  rivets. 

Many  of  the  preceding  experimental  results  were  obtained 
from  machine  work. 


Art.  74.— Welded  Joints. 

At  the  present  time  the  process  of  welding  can,  with  proper 
care  and  material,  be  made  to  give  excellent  results. 

Scarf  welds  give  much  better  results  than  lap  welds,  on 
account  of  the  bending  to  which  the  latter  are  subjected. 

Mr.  Kirtley  (Institute  of  Mechanical  Engineers  of  Great 
Britain)  made  some  experiments  with  small  strips,  7.5  inches 
long  and  ^  inch  thick,  cut  across  welded  joints,  These  strips 


644 


PIN  CONNECTION. 


Art.  75.] 


were  taken  out  of  boilers  whose  longitudinal  joints  had  been 
welded.  Twenty-three  experiments  with  strips  varying  from 
one  to  one  and  a  half  inches  wide,  gave  the  following  results 
per  square  inch  of  plate  section  : 


SOLID    PLATE. 


Greatest 53,3io  Ibs 57,79°  Ibs. 

Mean 46,140   "    52,860   " 

Least 36,960   "   46,370  " 

The  mean  result  is  seen  to  be  nea'rly  90  per  cent,  of  that 
of  the  solid  plate. 

Although  the  test  specimens  were  altogether  too  small  to 
be  of  the  greatest  value,  the  results  are  most  excellent. 

The  value  of  a  welded  joint  depends  as  well  upon  the 
nature  of  the  material  welded  as  upon  the  manipulation  during 
welding. 

Art.  75. — Pin  Connection. 

A  pin  connection  consists  of  two  sets  of  eye  bars  or  links, 
through  the  heads  at  one  end  of  each  of  which  a  single  pin 
passes.  Fig.  i  shows  a  pin  connection;  A,  A,  B,  B,  are  eye 
bars  or  links,  and  P  is  the  pin. 


rr 


Fig.  1 


The  head  of  the  eye  bar  (one  is  shown  in  elevation  in  Fig. 
2)  requires  the  greatest  care  in  its  formation.     It  is  imperfect 


Art.  75.] 


EYE  BAR  HEADS. 


645 


unless  it  be  so  proportioned  that  when  the  eye  bar  is  tested  to 
failure,  fracture  will  be  as  likely  to  take  place  in  the  body  of 
the  bar  as  in  the  head — in  other  words,  unless  its  efficiency  is 
unity. 

In  Fig.  2  the  head  of  the  eye  bar,  or  link,  is  supposed  to  be 
of  the  same  thickness  as  that  of  the  body  of  the  bar  whose 
width  is  w. 


F   N    D 


I 

\ 

Fig.  2 

If  /  is  the  thickness  of  the  bar,  so  that  wt  is  the  area  of  its 
normal  section,  then  t  is  almost  invariably  included  between 
the  limits  of  l/^w  and  l/&w.  In  fact  these  extreme  values  are 
each  too  extreme  for  the  full  resistance  of  the  bar,  although 
they  are  sometimes  used.  These  ratios,  as  well  as  the  diameter 
of  the  pin  in  terms  of  w,  can  only  be  determined  by  experi- 
ments on  full-sized  bars.  A  large  number  of  such  experiments 
have  been  made  both  in  this  country  and  in  Great  Britain,  and 
while  the  resistance  of  the  bar  as  a  whole  depends  to  a  con- 
siderable extent  on  the  mode  of  manufacture  or  formation  of 
the  head,  it  has  been  found  that  for  the  best  proportioned  head 
/  should  range  from  ]/±w  to  ^ze>,  and  the  diameter,  d,  of  the 
pin  from  0.752^  to  w. 

It  is  extremely  difficult  to  reach  more  than  a  general  idea 
of  the  condition  of  stress  in  an  eye  bar  head,  although  an  ap- 
proximate mathematical  treatment  of  the  question  may  be 
found  in  the  "Trans.  Am.  Soc.  of  Civ.  Engrs.,"  Vol.  VI.,  1877, 


646  PIN  CONNECTION.  [Art.  75. 

in  which  the  results  agree  essentially  with  those  of  experi- 
ment. 

Before  taking  a  general  view  of  the  stresses  which  may 
arise  in  an  eye  bar  head,  it  must  be  premised  that  a  difference 
of  -^V  to  roV'  between  the  diameter  of  the  pin  and  that  of  the 
pin-hole  is  exceptionally  good  practice.  Before  the  eye  bar  is 
strained,  therefore,  there  is  a  line  of  contact  only  between  the 
pin  and  eye  bar  head,  but  on  account  of  the  elasticity  of  the 
material,  this  line  changes  to  a  surface  when  the  bar  is  under 
stress,  and  increases  with  the  degree  of  stress  to  which  the  bar 
is  subjected.  The  line  and  surface  of  contact  is,  of  course,  in 
the  vicinity  of  Q,  Fig.  2,  i.e.,  on  that  side  of  the  pin  toward 
the  nearest  end  of  the  bar.  The  consequence  of  this  is,  that 
when  the  bar  is  strained,  the  portion  about  QB,  Fig.  2,  is  sub- 
ject to  direct  compression  and  extension  ;  that  about  RL,  NE 
and  GS  to  direct  tension  and  bendhig,  while  in  the  vicinity  of 
T  there  is  a  point  of  contra- flexure,  and  the  stress  in  the  direc- 
tion of  the  circumference  changes  from  compression  to  tension 
as  E  is  approached  from  Q. 

As  a  result  of  many  of  the  experiments  which  have  been 
made,  the  following  mode  of  proportioning  the  head  has  here- 
tofore been  very  extensively  used  :  Let  r  represent  the  radius 
of  the  pin,  while  reference  is  made  to  Fig.  2.  Then  take 
EN  =  o.66w.  The  curve  DRBK  is  a  semicircle  with  a  radius 
equal  to  r  -f  o.66w,  with  a  centre,  Ay  so  taken  on  the  centre 
line  of  the  bar  that  QB  =  0.87^.  GF  is  a  portion  of  the  same 
curve,  with  A'  as  the  centre  (A'C  =  AC);  GH  is  any  curve 
with  a  long  radius  joining  GF  gradually  with  the  body  of  the 
bar.  HG  should  be  very  gradual  in  order  that  there  may  be  a 
large  amount  of  metal  in  the  vicinity  of  CG,  for  there  the 
metal  is  subjected  to -flexure  as  well  as  direct  tension.  FD 
is  a  straight  line  parallel  to  the  centre  line  of  the  bar. 

As  the  preceding  rule  gives  a  head  whose  outline  causes  a 
more  expensive  die  than  a  simple  circle,  at  the  present  day 
eye  bar  heads  are  usually  formed  as  shown  in  Fig.  3. 


Art  75.] 


EYE  BAR  HF.ADS. 


Fig.3 


ABD  is  a  semicircle  with  a  radius  equal  to  r  -j-  o.8w  to 
r  +  0.9^,  and  whose  centre  C  is  the  centre  of  the  pin-hole. 
The  portions  FA  and  HD  are  formed  as  before. 

There  should  be  no  weld  across  the  bar  in  the  vicinity 
of  FH.  Consequently,  heads  are  usually  formed  by  placing 
proper  sized  pieces  upon  the  upset  ends  of  the  plain  bars,  and 
then,  after  insertion  in  a  heating  furnace,  forcing  the  head  to 
the  desired  shape  in  a  die  under  hydraulic  or  steam  pressure. 

The  intensity  of  this  pressure  will  affect,  to  a  considerable 
extent,  the  permissible  dimensions  of  the  head.  The  greater 
the  pressure,  the  better  will  be  the  results. 

The  unfinished  head  is  sometimes  rolled  on  the  bar,  as  by 
the  Kloman  process. 

The  thickness  of  the  head  is  sometimes  made  greater  than 
that  of  the  body  of  the  bar.  If  the  head  is  circular,  as  in  Fig. 
3,  the  section  of  metal  on  each  side  of  the  pin  (through  A  C  or 
CD]  should  be  not  far  from  eight-tenths  that  in  the  body  of 
the  bar. 

This  thickening  of  the  eye  bar  head  is  an  excellent  thing 
for  the  bar,  but  subjects  the  pin  to  a  great  increase  of  bending, 
and  in  that  respect  very  injurious. 

In  pin  connections,  the  pin  is  subjected  to  very  heavy 
bending.* 


*  For  a  detailed  treatment  of  this  subject,  the  author's  "Bridge  and  Roof 
Trusses  "  may  be  consulted. 


648  CABLES  OR  ROPES.  [Art.  76. 

If  M  is  the  bending  moment  to  which  the  pin  is  subjected, 
K  the  greatest  intensity  of  bending  stress  developed,  and  A 
the  area  of  the  normal  section  of  the  pin,  Eq.  (4)  of  Art.  63 
gives  : 


M=  K         =  o.iKd*  (nearly)     .     .     .     .     (i) 

o 

Or: 


.......      (2) 

Values  of  K,  for  circular  sections,  may  be  found  in  Art.  63. 


Art.  76. — Iron,  Steel  and  Hemp  Cables  or  Ropes. — Wrought-Iron  Chain 

Cables. 

The  following  tables  of  resistance  and  other  properties  of 
cables  are  those  published  by  John  A.  Roebling's  Sons  Co. 

It  will  be  observed  that  the  figures  for  hemp  ropes  are 
given  in  comparison  with  either  iron  or  steel  in  each  of  the 
tables. 

In  considering  the  resistance  of  iron  and  steel  cables  com- 
posed of  wire  twisted  into  strands,  it  is  of  the  highest  impor- 
tance to  keep  clearly  in  view  the  circumstances  or  conditions 
produced  by  the  manner  of  fabrication,  as  they  are  peculiar  to 
all  classes  of  ropes,  whether  of  hemp  or  wire. 

In  this  class  of  material  the  fibres  or  strands  no  longer  lie 
parallel  to  the  direction  of  the  stress  which  they  carry,  but  the 
process  of  twisting  causes  each  fibre  or  wire  to  take  a  helical 
form,  the  pitch  of  which  is  not  constant  for  the  different  por- 
tions of  the  rope.  The  consequence  is  that  if  the  process  of 
fabrication  were  absolutely  perfect,  so  that  each  wire  or  fibre 
could  take  its  proper  portion  of  load,  the  stress  in  that  wire  or 
fibre  would  be  its  portion  of  load  multiplied  by  the  secant  of 


Art.  76.] 


CABLES  OR  ROPES. 


649 


its  inclination  to  the  axis  of  the  rope.  As  a  matter  of  fact, 
however,  each  wire  does  not  take  its  proper  portion  of  load  ; 
the  imperfections  unavoidably  incident  to  the  processes  of 
manufacture  render  such  a  result  impossible.  Hence  the  in- 
creased necessity  of  experimental  determination  of  the  ulti- 
mate resistances  of  metallic  and  hemp  ropes. 

The  same  composite  character  of  these  productions  renders 
anything  like  an  approximately  elastic  character,  even,  an 
essential  impossibility.  It  is  true  that  any  rope  will  yield  to  a 
considerable  extent  while  under  stress,  and  then  return  nearly 
to  its  original  condition,  but  this  behavior  is  only  apparently 
elastic ;  it  is  almost  entirely  due  to  the  increase  of  helical  pitch 
of  the  strands  caused  by  the  external  loading.  During  this 

Standard  Hoisting  Ropes  with  19    Wires  to  the  Strand. 


. 

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CABLES  OR  ROPES. 


[Art.  76. 


Standard  Hoisting  Ropes  with  19   Wires  to  the  Strand. 


CAST  STEEL. 


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operation  the  strands  endeavor  to  place  themselves  more 
nearly  parallel  to  the  direction  of  stress,  and  give  rise  to  a  cor- 
responding decrease  in  diameter.  Since  these  influences  pre- 
clude the  existence  of  either  coefficient  of  elasticity  or  elastic 
limit,  ultimate  resistances  only  will  be  given  in  this  section. 

The  preceding  observations  evidently  do  not  apply  to  sus- 
pension bridge  cables  which  are  built  up  of  parallel  wires.  The 
operations  leading  to  the  production  of  such  a  cable  are  of 
such  a  refined  and  exact  character  that  the  total  resistance  of 
the  cable  may  be  assumed  without  essential  error  to  be  the 
sum  of  the  resistances  of  all  the  wires  taken  separately  :  the 
coefficient  of  elasticity  and  elastic  limit  may,  and  usually  do 
exist  with  perfect  definition. 


Art.  76.] 


CABLES  OR  ROPES. 


Galvanized  Steel  Cables  for  Suspension  Bridges. 


ULTIMATE   STRENGTH    IN   TONS    OF 

DIAMETER    IN   INCHES. 

WEIGHT    PER  FOOT. 

2,OOO    POUNDS. 

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652 


CHAIN  CABLES. 


[Art.  76. 


Transmission  and  Standing  Ropes  with  7    Wires  to  the  Strand. 


CAST  STEEL. 


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34 

Wrought-Iron  Chain  Cables. 

It  might  at  first  sight  be  supposed  that  the  pull  which  the 
link  of  a  chain  cable  could  resist  would  be  twice  that  offered 
by  a  bar  of  round  iron  equal  in  cross  section  to  that  of  one  side 
of  the  link.  But  a  weld  exists  at  one  end  of  the  link  and  a 
bend  at  the  other,  each  requiring  at  least  one  heat  for  the  por- 
tion of  the  link  in  which  it  is  located.  These  manipulations 
produce  a  considerable  decrease  in  the  resistance  of  the  link. 

The  United  States  Committee  on'"  Tests  of  Chain  Cables," 
of  which  Commander  L.  A.  Beardsley  was  chairman,  made 
many  experiments  on  the  iron  of  which  chain  cables  are  made, 
as  well  as  on  the  finished  cables. 


Art.  76.] 


CHAIN  CABLES. 


653 


The  following  conclusions  and  table  are  taken  from  the 
report  of  that  committee :  "  .  .  .  that  beyond  doubt,  when 
made  of  American  bar  iron,  with  cast-iron  studs,  the  studded 
link  is  inferior  in  strength  to  the  unstudded  one. 


Ultimate  Resistance  and  Proof  Tests  of  Chain   Cables. 


AV.   RESIST.   = 

PROOF 

AV.    RESIST.   = 

PROOF 

DIAM.  OF  BAR. 

DIAM.  OF  BAR. 

163^0   OF   BAR. 

TEST. 

163^   OF   BAR. 

TEST. 

Inches. 

Pounds. 

Pounds. 

Inches. 

Pounds. 

Pounds. 

I 

71,172 

33,840      \ 

i-rV 

162,283 

77,159 

Il1ff 

79,544 

37,820 

174,475 

82,956 

88,445 

42,053 

jH 

187,075 

88,947 

97,731 

46,468 

I  3 

200,074 

95,128 

I  ^ 

107,440 

51,084 

III 

213,475 

101,499 

i  A 

"7,577 

55,903 

ll 

227,271 

108,058 

if 

128,129 

60,920 

III 

241,463 

114,806 

*A 

139,103 

66,138 

2 

256,040 

121,737 

I| 

150,485 

71,550         ! 

"  That,  when  proper  care  is  exercised  in  the  selection  of 
material,  a  variation  of  five  to  seventeen  per  cent,  of  the 
strongest  may  be  expected  in  the  resistance  of  cables.  With- 
out this  care,  the  variation  may  rise  to  twenty-five  per  cent. 

"  That  with  proper  material  and  construction  the  ultimate 
resistance  of  the  chain  may  be  expected  to  vary  from  155  to 
170  per  cent,  of  that  of  the  bar  used  in  making  the  links, 
and  show  an  average  of  about  163  per  cent. 

"  That  the  proof  test  of  a  chain  cable  should  be  about  50  per 
cent,  of  the  ultimate  resistance  of  the  weakest  link." 

The  decrease  of  the  resistance  of  the  studded  below  the 
unstudded  cable  is  probably  due  to  the  fact  that  in  the  former 
the  sides  of  the  link  do  not  remain  parallel  to  each  other  up  to 
failure,  as  they  do  in  the  latter.  The  result  is  an  increase  of 
stress  in  the  studded  link  over  the  unstudded  in  the  proportion 


654  CHAIN  CABLES.  [Art.  76. 

of  unity  to  the  secant  of  half  the  inclination  of  the  sides  of  the 
former  to  each  other. 

From  a  great  number  of  tests  of  bars  and  finished  cables, 
the  committee  considered  that  the  average  ultimate  resistance, 
and  proof  tests  of  chain  cables  made  of  the  bars,  whose  diam- 
eters are  given,  should  be  such  as  are  shown  in  the  accompany- 
ing table. 


CHAPTER    XI. 

MISCELLANEOUS  PROBLEMS. 

Art.  77.  —  Resistance  of  Flues  to  Collapse. 

IF  a  circular  tube  or  flue  be  subjected  to  external  normal 
pressure,  such  as  that  of  steam  or  water,  the  material  of  which 
it  is  made  will  be  subjected  to  compression  around  the  tube,  in 
a  plane  normal  to  its  axis.  If  the  following  notation  be 
adopted  : 

/  =  length  of  tube  ; 

d  =  diameter  of  tube  ; 

/  =  thickness  of  wall  of  the  tube  ; 

/  =  intensity  of  excess  of  external  pressure  over  internal, 

then  will  any  longitudinal  section  //,  of  one  side  of  the  tube,  be 
subjected  to  the  pressure  —  —  .  But  let  a  unit  only  of  length 

of  tube  be  considered.  This  portion  of  the  tube  is  approxi- 
mately in  the  condition  of  a  column  whose  length  and  cross 
section,  respectively,  are  nd  and  /. 

The  ultimate  resistance  of  such  a  column  is,  Art.  25  : 


P= 


n2d2    ' 
As  this  ideal  column  is  of  rectangular  section  : 


656  COLLAPSE    OF  FLUES.  [Art.  77. 


'-£• 


and 


But  P  =  pd,  hence  : 

(0 


is  the  greatest  intensity  of  external  pressure  which  the  tube 
can  carry.  But  the  formulae  of  Art.  25  are  not  strictly  applicable 
to  this  ideal  column.  The  curvature  on  the  one  hand  and  the 
pressure  on  the  other  tend  to  keep  it  in  position  long  after  it 
would  fail  as  a  column  without  lateral  support.  Hence,  p  will 
vary  inversely  as  some  power  of  d  much  less  than  the  third. 

Again,  it  is  clear  that  a  very  long  tube  will  be  much  more 
apt  to  collapse  at  its  middle  portion  than  a  short  one,. as  the  latter 
will  derive  more  support  from  the  end  attachments ;  and  this 
result  has  been  established  by  many  experiments.  Hence,/ 
must  be  considered  as  some  inverse  function  of  the  length  /. 

Eq.  (i),  therefore,  can  only  be  taken  as  typical  in  form,  and 
as  showing  in  a  general  way,  only,  how  the  variable  elements 
enter  the  value  of/.  If  x,  y  and  z,  therefore,  are  variable  ex- 
ponents to  be  determined  by  experiment,  there  may  be  written  : 


=  c 


Pd* 


in  which  c  is  an  empirical  coefficient. 

Sir  Wm.  Fairbairn  ("  Useful  Information  for  Engineers, 
Second  Series  ")  made  many  experiments  on  wrought-iron  tubes 
with  lap  and  butt  joints  single  riveted.  He  inferred  from  his 


Art.  77.]  COLLAPSE   OF  FLUES. 

tests  that  y  =  z  =  I.     Two  different  experiments  would  then 
give: 

pld=.ct*      .      .     .     ...      .     (3) 


Hence, 

log  (fid)  =  log  c  +  x  log  t  ; 

log  (p'l'd')  =  log  c  +  x  log  t'  ; 

in  which  "log"  means  "  logarithm."     Subtracting  one  of  these 
last  equations  from  the  other,  the  value  of  x  becomes  : 


x  =  log(pld)  -  log  (p •Id')  = 
log  t  —  log  t' 


As  /,  /,  d,  t,  p',  /',  d'  and  /'  are  known  numerical  quantities 
in  every  pair  of  tests,  x  can  at  once  be  computed  by  Eq.  (5) ; 
c  then  immediately  results  from  either  Eq.  (3)  or  Eq.  (4).  By 
the  application  of  these  equations  to  his  experimental  data, 
Fairbairn  found  for  wrought-iron  tubes  : 

/  =  9,675,600  (6) 


in  which  /  is  in  pounds  per  square  inch,  while  t,  I  and  d  are  in 
inches.  Eq.  (6)  is  only  to  be  applied  to  lengths  between  18  and 
1 20  inches. 

He  also  found  that  the  following  formula  gave  results  agree- 
ing more  nearly  with    those  of  experiment,  though  it  is  less 

simple  : 

42 


658  COLLAPSE   OF  FLUES.  [Art.  77. 

/2-19  d 

P  =  9>675,&>o  -ft  -  0.002  -  ......    (7) 

Fairbairn  found  that  by  encircling  the  tubes  with  stiff  rings 
he  increased  their  resistance  to  collapse.  In  cases  where  such 
rings  exist,  it  is  only  necessary  to  take  for  I  the  distance  between 
two  adjacent  ones. 

In  1875  Prof.  Unwin,  who  was  Fairbairn's  assistant  in  his 
experimental  work,  established  formulae  with  other  exponents 
and  coefficients  ("  Proc.  Inst.  of  Civ.  Engrs.,"  Vol.  XLVL). 
He  considered  x,  y  and  z  variable,  and  found  for  tubes  with  a 
longitudinal  lap  joint  : 


P  =  7,363,000  j  ......    (8) 

From  one  tube  with  a  longitudinal  butt  joint,  he  deduced  : 

/2-21 
/  =  9,614,000  j^—t    ......     (9) 


For  five  tubes  with  longitudinal  and  circumferential  joints, 
he  found  : 

/=  15,547,000— -6 (10) 


By  using  these  same  experiments  of  Fairbairn,  other  writers 
have  deduced  other  formulae,  which,  however,  are  of  the  same 
general  form  as  those  given  above.  It  is  probable  that  the 
following,  which  was  deduced  by  J.  W.  Nystrom,  will  give 
more  satisfactory  results  than  any  other : 

/  =  692,800^ (ii) 


Art.  78.]  SOLID  ROLLERS.  659 

At  the  same  time,  it  has  the  great  merit  of  more  simple 
application. 

'From  one  experiment  on  an  elliptical  tube,  by  Fairbairn,  it 
would  appear  that  the  formulae  just  given  can  be  approximately 
applied  to  such  tubes  by  substituting  for  d,  twice  the  radius  of 
curvature  of  the  elliptical  section  at  either  extremity  of  the 
smaller  axis.  If  the  greater  diameter  or  axis  of  the  ellipse  is  #, 

and  the  less  b  ;  then,  for  d,  there  is  to  be  substituted  -=-  . 


Art.  78. — Approximate  Treatment  of  Solid  Metallic  Rollers. 

An  approximate  expression  for  the  resistance  of  a  roller 
may  easily  be  written.  The  approximation  may  be  considered 
a  loose  one,  but  it  furnishes  a  basis  for  an  accurate  empirical 
formula. 

The  roller  will  be  assumed  tc  be  composed  of  indefinitely 
thin  vertical  slices  parallel  to  its  axis.  It  will  also  be  assumed 
that  the  layers  or  slices  act  independently  of  each  other,  and, 
finally,  that  the  material  above  the  roller  is  of  a  thickness  equal 
to  its  radius. 

In  Fig.  i  let  AC  —  d. 
In  Fig.  i  let  DC  =  e. 

Let  E'  =  coefficient  of  elasticity  for  the  material  over 

the  roller. 
Let  E  =  coefficient  of  elasticity  for  the  material  of  the 

roller. 

Let  w  =  intensity  of  pressure  at  A. 
Let  P   =  total  weight  which  the  roller  sustains  per  unit 

of  length. 


66o 


CYLINDRICAL  ROLLERS. 


[Art.  78. 


Fig.1 


Let  x  and  y  be  measured 
horizontally  and  vertically, 
respectively,  from  A  as  the 
origin. 

From  Fig.  I  : 

BC=j?,AB\     and 

i^-t^r  .  .  (,) 


If  /   is   the    intensity    of 
vertical  pressure  at  any  point, 


^,         A'B'       , 
pdx  =  —  wdx 


(2) 


But  by  Eq.  (i)  : 


7A'C'. 


E  +  E 
Also,  if  R  is  the  radius  of  the  roller : 


A'Cr=d—  y\     and 


w 

'£' 


Hence, 


From  the  equation  of  the  circle 


y  =  R  - 


Art.  78.]  CYLINDRICAL  ROLLERS.  66l 


Since 

P  =  2  r<//>  there  results  : 

•'o 


Eq.  (4)  can  be  very  much  simplified  for  all  ordinary  cases. 
From  what  has  preceded  : 


When  ^  is  small  :     sin-'1  -75-  =  -15-  ;     ^  =  ^2Rd  -f  df2  ;   and 

.A.        /L 

-  ra)K  =  R  -  ^  ,  nearly. 
Substituting  in  Eq.  (4): 


P  - 


Hence,  as  d  is  small,  nearly  : 
Or,  for  length  / : 


A  simple  expression  for  conical  rollers  may  be  obtained  by 
using  Eq.  (6). 


662 


CONICAL  ROLLERS. 


[Art.  78. 


As  shown  in  Fig.  2,  let  z  be  the  distance  parallel  to  the 
axis  of  any  section  from  the  apex  of  the  cone  ;  then  consider  a 


Fig.2 


portion  of  the  conical  roller  whose  length  is  dz.  Let  RT  be  the 
radius  of  the  base.  The  radius  of  the  section  under  consider- 
ation will  then  be  : 


and  the  weight  it  will  sustain  : 


Hence, 


EE 


.     .     .     (8) 


Eqs.  (7)  and  (8)  give  ultimate  resistances  if  w  is  the  ultimate 
intensity  of  resistance  for  the  roller. 

It  is  to  be  observed  that  the  assumption  on  which  the  in- 
vestigation is  based  leads  to  an  error  on  the  side  of  safety. 


Art.  79.] 


SPIKE  DRIVING  AND  DRAWING. 


663 


Art.  79. — Resistance  to  Driving  and  Drawing  Spikes. 

Some  very  interesting  experiments  on  driving  and  drawing 
rail  spikes  were  made  by  Mr.  A.  M.  Wellington,  C.  E.,  and  re- 
ported by  him  in  the  "  R.  R.  Gazette,"  Dec.  17,  1880.  He  ex- 
perimented with  wood  both  in  the  natural  state  and  after  it 
had  been  treated  by  the  Thilmeny  (sulphate  of  baryta)  pre- 
serving process. 

"  The  test  blocks  were  reduced  to  a  uniform  thickness  of  4.5 
inches  ;  this  thickness  being  just  sufficient  to  give  a  full  bear- 
ing surface  to  the  parallel  sides  of  the  spikes  when  driven  to 
the  usual  depth,  and  to  allow  the  point  of  the  spike  to  project 
outwards.  It  was  considered  that  the  beveled  point  could  add 

Spikes  were  Standard :  5.5  inches  x  -&  inch. 


KIND   OF  WOOD. 


NATURAL    WOOD. 


To  driving 
spike,  pounds. 


To  pulling 
spike,  pounds. 


PREPARED    WOOD. 


To  driving 
spike,  pounds. 


To  pulling 
spike,  pounds. 


Beech  

White  oak,  green 

Pin  oak 

White  ash 

White  oak,  well  seasoned. 


Mean. 


5,953 


Black  ash 


Elm 

Chestnut,  green, 
Soft  maple 

Sycamore 

Hemlock  . . . 


2,910 


5,673 
6,282 


6,638 
6,469 


5,^8 
3,435 

4,408 
4,868 

3,536 
3,843 
2,730 
3790 
2,578 

3'6& 
3,188 

3,i88 


Mean. 

[  5,978 
[  6,523 
[6,553 
4,560 
[4,281 

4,638 

3,69° 
3,260 

3,"* 
3,188 
1,996 


Mean. 


4,453 


Mean. 

IS?}— 


(Split.) 

3-340 

8S2 

3,493 


1,968 


664  SHEARING  BY  BOLTS  AND  KEYS.  [Art.  80. 

very  little  to  the  holding  power  of  the  spike,  and  it  was  desired 
to  press  the  spike  out  again  by  direct  pressure  after  turning  the 
block  over. 

The  forces  exerted  in  pulling  and  driving  the  spikes  were 
produced  by  a  lever.  A  few  tests  with  a  hydraulic  press 
showed  that  the  friction  of  the  plunger  varied  from  about  6  to 
1 8  per  cent. 

The  accompanying  table  gives  the  results  of  the  experi- 
ments. 


Art.  80. — Shearing  Resistance   of  Timber  behind    Bolt   or   Mortise 

Holes. 

Col.  T.  T.  S.  Laidley,  U.S.A.,  made  some  tests  during  1881 
at  the  United  States  Arsenal,  Watertown,  Mass.,  on  the  resist- 
ance offered  by  timber  to  the  shearing  out  of  bolts  or  keys, 
when  the  force  is  exerted  parallel  to  the  fibres. 


Fig.1 


Fig,2 


The  test  specimens  are  shown  in  Figs.  I  and  2.  Wrought- 
iron  bolts  and  square  wrought-iron  keys  were  used.  All  the 
timber  specimens  were  six  inches  wide  and  two  inches  thick. 
The  diameter  of  the  bolts  used  (Fig.  i)  was  one  inch  for  all  the 
specimens.  The  keys  were  i"  x  1.5"  and  1.125"  x  1.5"  as 
shown  in  Fig.  2.  In  all  the  latter  specimens,  failure  took  place 
in  front  of  the  smaller  key  where  the  pressure  was  greatest. 

In  many  cases  the  specimen  sheared  and  split  simultane- 
ously in  front  of  the  hole.  By  putting  bolts  through  the 
pieces  in  a  direction  normal  to  the  force  exerted,  so  as  to  pre- 


Art.  8 1.] 


BULGING   OF  PLATES. 


665 


vent  splitting,  the  resistance  was  found  (in  most  cases)  to  be 
considerably,  though  irregularly  increased. 


CENTRE  OF 

HOLE    FROM 

TOTAL   AREA 

ULTIMATE   SHEARING  RESISTANCE   PER 

KIND    OF    WOOD. 

END    OF 

OF  SHEARING. 

SQUARE   INCH,   IN   POUNDS. 

SPECIMEN. 

Inches. 

Sq.  inches. 

2 

8 

399 

Spruce  (bolts) 

.      4 
6 

16 
24 

359 
275 

8 

'    3o 

202 

White  pine  (bolts)  

2 

.     4 
6 

8 
16 

24 

457 
611 

45° 

8 

32 

'  I27 

(    2 

1     8 

607 

Yellow  pine  (bolts)  

1 

J    z6 
)    24 

720 
456 

\  8 

'    3? 

337 

(    2 

B 

5Q9 

Yellow  pine  (square  keys)  .  . 

< 

16 
24 

369 

572 

(    7 

28 

,  438 

White  pine  (square  keys)  .  .  . 

8 
16 
24 
28 

550 
412 

332 

216 

Spruce  (square  keys)  

i 

8 
16 
24 

(  410  (not  thorough\y  seasoned.) 
"j    242  (wet  timber). 

(   7 

28 

C  279 

Unless  otherwise  stated,  the  wood  was  thoroughly  sea- 
soned. 

The  accompanying  table  gives  the  results  of  Col.  Laidley's 
tests. 


Art.  81.— Bulging  of  Plates. 

A  plate  offers  resistance  to  "  bulging "  when  it  is  simply 
supported,  or  firmly  fixed,  around  its  entire  edge,  and  carries 
a  single,  or  uniformly  distributed  load  acting  normal  to  its 
surface.  The  very  complicated  nature  of  the  stresses  and 
strains  existing  in  a  plate  thus  acted  upon,  together  with  the 
fact  that  its  conditions  just  before  rupture  are  entirely  different 


666  BULGING  OF  PLATES.  [Art.  8  1. 

from  those  accompanying  the  initial  loading,  give  to  the  prob- 
lem a  character  of  unusual  intricacy,  and,  indeed,  preclude  a 
solution  possessing  a  degree  of  approximation  commonly  ob- 
tained in  questions  relating  to  the  elasticity  and  resistance  of 
materials. 

An  elegant  analysis  of  the  problem,  considered  as  one  of 
pure  elasticity,  may  be  found  in  "  Die  Theorie  der  Elasticitat 
Fester  Korper,"  by  Clebsch.  It  is,  however,  of  little  value  in 
connection  with  questions  of  ultimate  resistance. 

The  following  roughly  approximate,  but  simple,  analysis  may 
be  used  to  suggest  the  form  of  an  empirical  formula  which  can 
be  completed  by  the  aid  of  experiments. 

Let  the  length,  breadth  and  thickness  of  a  rectangular 
plate  simply  supported  around  its  edges,  be  represented  by  a, 
b  and  /,  respectively,  and  let  it  first  be  loaded  by  a  uniformly 
distributed  pressure  whose  intensity  (per  unit  of  aft)  is  w. 

If  the  plate  is  supposed  to  consist  of  two  sets  of  small 
strips  or  beams  parallel  to  a  and  b,  those  crossing  each  other 
at  the  centre  must  have  the  same  deflection  at  middle.  If, 
further,  the  uniform  load  w  be  supposed  to  be  so  divided  into 
two  parts,  wl  and  w\  that  they  would  cause  two  rectangular 
beams  whose  spans  are  a  and  b  to  have  the  same  centre  deflec- 
tion, the  following  equation  (see  Eq.  (26)  of  Art.  24)  must 
obtain  : 


Then,  since  w'  -f-  wt  =  w,  there  must  result  : 

;    and    «'i  •= 


The  bending  moments  at  the  centres  of  such  beams  would 
be  (Eq.  (27),  Art.  24,  and  Eq.  (14),  Art.  18)  : 


Art.  8  1.]  FORMULAE.  667 


iv.a2  _    2KJ  w'b*       2KI 

______;     and     —  =  -_  . 


Since  the  beams  are  rectangular  in  section,  /  =  —  . 
Hence  : 


;  =  ;     and  K>  =-. 

4'*  4*2 


According  to  these  hypothetical  conditions  the  greatest  in- 
tensity of  stress  at  the  centre  of  the  plate  will  have  the  value  : 


K 
Hence : 


A         2         )  A**- 


For  square  plates,  a  —  b. 
Hence 


v       3a*w  ^  fw 

^-  ;     and     /  =  0.615^   ./—     .     .     . 


(3) 


If  the  edges  are  fixed,  the  greatest  bending  will  occur  along 
those  lines  ;  and  for  Kv  and  K'  then  are  to  be  put  y^Kl  and 
'. 
Hence  : 

,      r,, 

and       -^ 


Since  the  greatest  bending  occurs  along  the  edges,  these 


668  BULGING  OF  PLATES.  [Art.  8 1. 

are  the  expressions  for  the  greatest  intensities  of  stress.     If  ab* 
is  greater  than  a2b,  then  is  K^  greater  than  K' ;  and  vice  versa. 
In  the  first  case  the  expression  for  /  is : 


°-7°7a*  '      +  »)  *.  •  • 


But  if  K'  >  KV    or,    a*b  >  ab2  : 


v   V»«  +  *)A-'    •     •     •     •    (6) 
If  the  plate  is  square  : 


If  a  plate  is  loaded  with  a  single  weight  P,  it  may  be  sup- 
posed to  be  divided  in  the  same  manner  as  w ;  so  that 

/>/>'  =  P. 


The   equation   of  middle   deflections  for  ends  simply  sup- 
ported then  becomes  : 


.       =    P'P 
Hence : 


Proceeding  in  precisely  the  same  manner  as  before : 


Art.  8 1.]  FORMULA.  669 

abP 


+  *        ....     (8) 
\      •    ~  / 

and 

f  *J,V  \1A 

....     (9) 


If  the  plate  is  square  : 


~;    and,    t  =  0.87  ,,  .     .     (10) 


T/"  /^  edges  are  fixed  in  position,  the  hypothetical  beams 
are  fixed  at  each  end  and  loaded  at  the  centre,  and  the  greatest 
bending  moments  (at  centre  and  ends  alike)  are  thereby  re- 
duced to  one-half  their  preceding  values,  or,  what  is  the  same 
thing,  2t2  is  to  take  the  place  of  /2  in  Eqs.  (8),  (9)  and  (10). 

Hence : 

abP 

•K-    —    O.53     /2   /     3      ,        K 


If  the  plate  is  square  : 

r,  rt1/3  /rt/5 

^  =  0-375-^-;     and     /  =  0.613  \l -g     •     •     (13) 

These  equations  are  of  little  value  as  they  stand,  except  as 
indicating  a  form  of  formula  to  which  empirical  coefficients  are 
to  be  fitted.  The  hypothetical  division  of  the  plate  into  small 
beams  is  very  far  indeed  from  being  correct.  In  the  empirical 
determinations  which  follow,  therefore,  K  will  not  be  the 


670  BULGING  OF  PLATES.  [Art.  8 1. 

greatest  intensity  of  stress  in  the  plate,  but  a  coefficient  or 
quantity  partly  analytical  and  partly  experimental. 

Circular  plates  have  not  been  considered,  because  square 
ones  furnish  the  requisite  type  of  formula. 

Experiments  have  thus  far  been  made  on  square  and  circu- 
lar plates  only ;  hence,  oblong  rectangular  plates  will  not  again 
be  noticed. 

Kirkaldy's  experiments  on  Fagersta  steel  plates  and  Fair- 
bairn's  on  wrought-iron  ones  would  seem  to  indicate  that  the 
thickness  t  varies  about  as  (w)°-8  or  (P)°-8 ;  but  the  variation  in 
diameter  or  side  of  square  was  not  sufficient  to  establish  any 
relation  between  t  and  #,  while  other  elements  remain  the 
same.  Regarding,  therefore,  K  as  an  empirical  quantity  which 
may  have  different  values  for  square  and  circular  plates,  Eqs. 
(3),  (7),  (10)  and  (13),  may  be  written  as  follows  : 


and  t  =  0.6150  —=  .  .  (14) 
=  -^  iv*A  ;  and  /  =  0.50— =  ...  (15) 
-^P-6;  and  /  =  0.87  ^-^  .  .  .  (16) 


P«,     and     /  =  0.613  .     .      (17) 


Kirkaldy  made  twenty  experiments  with  mild  Fagersta 
steel  circular  plates,  12  inches  in  diameter.  He  forced  these 
through  an  aperture  10  inches  in  diameter  by  the  pressure  of  a 
very  blunt  point.  The  edge  of  the  aperture  on  which  the 
plate  rested  was  rounded  ;  hence  the  initial  diameter  of  aper- 


Art.  8 1.]  STEEL   CIRCULAR  PLATES.  6/1 

ture  was  somewhat  more  than  10  inches.  Eqs.  (16)  are  the 
ones  to  be  used  in  connection  with  these  experiments. 

From  the  first  member  of  that  equation,  K  was  computed 
for  a  number  of  different  experiments,  by  substituting  the 
numerical  values  of  P,  t  and  a.  In  this  manner  the  following 
values  were  found  to  give  good  results  : 

For  unannealed  mild  Fagcrsta  steel  circular  plates  : 

K  =  6,760,000,000. 
Hence  : 

/  —  0.000,010,6  VaP°* (18) 

For  annealed  mild  Fagersta  steel  circular  plates  : 

K  —  5,710,000,000. 
Hence  : 

/  =  0.000,011,52  ^~aP°*   .     ....     (19) 

Eq.  ( 1 6)  gives  : 


p  = 


Table  I.  contains  the  results  of  computation  by  this  formula 
and  those  obtained  in  the  tests.  On  account  of  the  rounded 
edge  of  the  supporting  ring,  K  was  so  taken  that  P,  as  com- 
puted, is  a  little  larger  than  its  experimental  value.  None 
of  these  plates  were  cracked,  but  they  were  bulged  at  the  cen- 
tre from  3.00  to  3.45  inches. 

In  "  Engineering"  for  Sept.  28,  1877,  Robert  Wilson  de- 
scribes four  experiments  on  unstayed  flat  boiler  heads  sub- 
jected to  hydraulic  pressure.  These  flat  circular  plates  were 


6/2 


BULGING  OF  PLATES. 


[Art.  8 1, 


TABLE   I. 
Circular  Plates  simply  Supported. 


UNANNEALED. 


P,  in  pounds. 

P,  in  pounds. 

Experimental. 

By  formula. 

Experimental. 

-Z?y  formula. 

1 

215,690 

219,420 

| 

198,000 

196,530 

\ 

162,740 

166,000 

1 

154,330 

148,690 

t 

104,850 

115,860 

a 

8 

95,600 

103,780 

* 

71,800 

69,800 

i 

59,430 

•  62,520 

i 

35,400 

29,350 

i 

25,430 

26,290 

Each  "experimental"  result  is  a  mean  of  two. 

riveted  to  angles  encircling  the  body  of  the  boiler.  The  edges 
of  the  plates  were  thus  fixed,  and  Eqs.  (15)  are  therefore  to  be 
used.  Proceeding  in  precisely  the  same  manner  as  before,  the 
following  values  were  established  : 

For  wr 'ought-iron  flat  boiler  heads,  with  fixed  edges  : 

K  —  11,000,000. 
Hence  : 

/  =  0.000,01  Srftt/0'8 (20) 

w  was  taken  in  pounds  per  square  inch ;  it  has  the  value  : 


Art.  8 1.]  WROUGHT-IRON  PLATES.  6/3 

The  results  of  the  experiments,  and  of  this  formula,  are  : 

DIAMETER,  t,  <—1U<  IN   POUNDS   PER   SQ.  IN.^, 

INCHES.  INCH.  Experimental,  By  formula. 


A-  

28o  

.     £  , 

26.  25.  . 

f  

,  371  

.    206 

28.2<:. 

.     ».. 

.    300.. 

.    270 

The  agreement,  in  this  case,  is  not  satisfactory.  It  is  prob- 
ably due  to  the  lack  of  a  proper  exponent  of  a.  These  plates 
were  fractured  along  the  lines  of  rivet  holes  in  the  edges. 

Two  means  of  four  experiments  by  Fairbairn  remain  to  be 
considered.  His  plates  were  square  ones  of  wrought  iron, 
firmly  fixed  to  a  square  frame  12  inches  by  12  inches  in  the 
aperture.  The  force  was  applied  by  a  blunt  point  at  the 
centre,  consequently  Eqs.  (17)  are  to  be  used. 

By  precisely  the  same  method  already  used,  the  following 
results  were  established  : 

For  wrought-iron  12-inch  square  plates,  with  edges  firmly 
fixed: 

K  =  390,000,000. 


t  =  0.000,031  VaP°s     .....     (21) 
The  expression  for  the  indenting  force  is  : 


The  experiments  and  computations  are  : 

DIAMETER,  /,  , /*,  IN   POUNDS. * 

INCHES.  INCH.  Experimental.  By  formula. 

12 \ 16,780 16,350 

12 £ 37,720 38>8QO 

The  plates  gave  way  at  the  centre,  under  the  blunt  point, 
43 


i'2^ 

674  SPECIAL   CASES  OF  FLEXURE.  [Art.  S2. 

Some  experiments  by  Kirkaldy,  in  1875,  on  wrought-iron 
circular  plates  simply  supported  around  the  edge,  show  that  for 
12-inch  plates  forced  through  a  lo-inch  aperture  with  rounded 
edge,  there  may  be  safely  taken  : 


t  —  0.000013  A/tf  P0-8 (22) 

In  all  the  preceding  formulae,  a  and  /  are  to  be  taken  in 
inches  ;  w  in  pounds  per  square  inch,  and  P  in  pounds. 

The  investigations  can  only  be  considered  provisional.  Al- 
though they  give,  as  a  whole,  tolerably  satisfactory  results,  the 
range  of  the  experiments  is  far  too  small  for  the  establishment 
of  thoroughly  reliable  formulae.  Experiments  on  which  a 
proper  exponent  of  a  can  be  based,  are  yet  wholly  lacking ;  and 
as  the  only  resort,  that  found  in  the  rough  analysis  has  been 
retained. 

Art.  82.— Special  Cases  of  Flexure. 

There  are  a  few  cases  of  flexure  which,  while  not  frequently 
found  in  engineering  experience^  are  of  some  practical  impor- 
tance, and  are  occasionally  required.  The  two  or  three  which 
follow  involve  the  integration  of  some  linear  differential  equa- 
tions that  are  treated  in  all  the  advanced  works  on  the  integral 
calculus ;  consequently  the  operations  of  integration  will  not 
be  given  here,  but  the  general  integrals  will  be  assumed. 

Flexure  by  Oblique  Forces 

In   Fig.   i    let   OX  represent  a  beam   acted   upon   by  the 

oblique  forces  P9  which 
make  angles  a  with  the 
axis  of  X.  The  origin  O 
is  supposed  to  be  taken 
x  anywhere  on  the  axis  of 


Fig.J  the  beam.     If  right-hand 


Art.  82.] 


SPECIAL  CASES  OF  FLEXURE. 


>75 


moments  are  positive  and  left-hand  negative,  the  component 
P  sin  a  will  have  the  negative  moment  —  P  sin  ax  about  O. 
The  lever  arm  of  P  cos  a,  if  the  deflection  w  is  positive,  is 
+  w,  and  its  moment  P  cos  a.  w  is  positive.  Hence  the  result- 
ant moment  of  any  force,  P,  in  reference  to  the  origin  O  is  : 

El  — —  =  —  P  sin  a  .  x  -f  P  cos  a  .  w     .     .     (i) 


If  a  is  greater  than  90°,  cos  a  is  negative,  so  that  if 
P  cos  a 


A  =  ± 


,     Tr 
and     F=  — 


a  •  x 


El  El 

the  two  cases  may  be  expressed  by  the  equation  : 


(2) 


If  a  =  -f  V  —  A,  and  b  —  —  V  —  A,  the  general   integral 
of  Eq.  (2)  is  : 


w 


=  Ceax  +  CV*H  --  <-—  \Ve-~dx  --  —~\Ve-lxdx',     (3) 
a  —  o  j  a  —  o  j 


in  which  C  and  Cl  are  arbitrary  constants  to  be  determined  by 
the  special  conditions  of  any  given  problem,  and  e  =  2.71828. 
When  a  is  less  than  90°  and 


Eq.  (3)  becomes  : 


=  Ce*x  +  C'e** 


i  .....     (4) 


6/6  SPECIAL   CASES  OF  FLEXUKE.  [Art.  82. 

C1,  is  another  arbitrary  constant  to  be  determined  by  the  par- 
ticular circumstances  of  a  given  case. 

The  conditions  on  which  the  determination  of  these  con- 
stants rest  are  expressed  by  giving  known  values  to  w  and  -z — 

for  values  of  x,  also  known. 
If  a  is  greater  than  90°  : 


.  y  —  i,     and     b  —  — 


and  Eq.  (3)  becomes  : 

f  „,     . 

^v  ==  Ccos  -      *+C  s^n     - 


x  —  tan  a  .  x  -f-  Cl     ......     (5) 

As  before,  C,  C'  and  £7,  are  to  be  determined  by  the  cir- 
cumstances of  each  case  to  which  the  equation  is  applied  ;  and 
the  value  of  cos  a,  it  is  to  be  remembered,  is  to  be  substituted 
with  the  positive  sign. 

Let  a  column  with  one  fixed  and  one  free  end,  and  with  the 
force  P  acting  parallel  to  its  original  axis,  be  considered.  Since 
a=  180°: 


w  =  C  cos 


Let  the  origin  of  co-ordinates  be  taken  at  the  free  end. 
Now  since  w  will  vary  from  o  to  its  greatest  value  at  the  fixed 
end,  there  can  be  no  portion  of  it  which  is  constant.  Hence, 
Ci  must  be  equal  to  zero.  Also,  w  must  equal  zero  for  x  —  o. 
This  condition  gives  C  =  o. 


Art.  82. J  COLUMNS.  677 

The  value  of  w  then  becomes  : 


p 

w  =  C  sin  {  \/  -       *    x  } (7) 


dw        r,      I  -  /       /  -  i  /ox 

.*.       -fa~^L\l~PT^os\\l~WT         *          •       •      • 


But  if  /  is  the  length  of  the  column,  -3—  =  o  for  x  =  I. 


Hence  : 


cos 


or  if  #  is  any  whole  number  from  o  to  infinity  : 

.  /  =  y2(2n  +  I)TT (9) 


f~P 

If  the  value  of  A  /  _  be  taken  from   Eq.  (o)  and  inserted 
V  El 

in  Eq.  (8),  there  will  result  : 
dw        r, 

•- 


Eq.  (10)  shows  that  for  values  of  ;tr  equal  to  I,  3,  5,  7    .    .    . 
times  —         -  ,   -——  =  o.     The  most  dangerous  supposition, 

2H  —  j—    I  CtfX  i 

i.  e.,  that  which  requires  the  greatest  value  of  P,  is  n  =  o. 


6/8  SPECIAL    CASES  OF  FLEXURE.  [Art.  82. 

This  value  of  n  in  Eq.  (9)  gives  : 


The  ultimate  resistance  of  the  column  is  thus  seen  to  be 
independent  of  the  deflection,  as  was  found  for  a  different  case 
in  Art.  25.  The  end  of  the  column,  in  this  case,  which  carries 
the  load  is  free  to  deflect  laterally,  but  in  Art.  25  both  ends 
were  supposed  to  be  fixed  in  a  lateral  direction  in  reference  to 
each  other.  In  the  latter  case  the  resistance  is  seen  to  be  nine 
times  as  great  as  in  the  present. 

Since  : 


C05\  \     -Wr  '  l     =o,    sin 


Hence,  if  «/x  is  the   deflection  of  the  free  end  from  a  vertical 
tangent  to  the  fixed,  Eq.  (7)  becomes,  for  x  =  I : 

w,  =  C. 
In  general,  therefore  : 


=  w    sin 


For  the  same  value  of  x,  therefore,  w  varies  directly  as  wv 
and  the  relative  deflections  may  be  computed  by  the  equa- 
tion : 


+ 

2l 


=  sn  (13) 


Art.  82.]  FLEXURE  BY  NORMAL  LOAD.  6/9 


or  in  the  ordinary  case  : 


w    .  7t,r  ,  x 

-  =  sin  — r- (14) 

w       2l 


Eq.  (i)  was  written  for  one  force  only.  If  any  number  of 
forces  act  : 

El  —7—  =  ^(  ~  P  sin  a  .  x  -f-  P  cos  a  .  w)  ; 
and  in  place  of  w  there  is  to  be  put  2w. 

General  Flexure  by  Continuous  Normal  Load. 

The  most  general  case  of  flexure  by  a  continuous  normal 
load,  is  that  in  which  the  intensity  (load  per  unit  of  length  of 
beam)  is  a  variable  quantity.  Let  x  be  an  abscissa  measured 
along  the  original  axis  of  the  beam,  and  let  w  represent  the 
deflection.  Then  the  intensity  of  the  load  may  be  represented 
by/(.r,  w).  It  was  shown  in  Art.  20  that  : 


The  integration  of  the  equation  : 

d*w         f(x,  w) 
~d^~  El      ' 

will  depend  upon  the  form  of  the  function  f(x,  w). 

Let  it  be  supposed  that  f(x,  w)  —  ex,  c  being  a  constant. 
Then  if  A,  Alt  A3  and  A3  are  constants  of  integration,  there 
will  result  : 


680  SPECIAL    CASES  OF  FLEXURE.  [Art.  82. 


Again,  if  f(x,  w)  =  cw,  c,  as  before,  being  a  constant  : 

d*w   _    cw^  ,     } 

~dx^       "El 

For  simplicity  of  notation,  let  : 


then  the  general  integral  of  Eq.  (17)  becomes  : 

w  =  Aeax  +  A,e-ax  +  A2  cos  ax  +  A3  sin  ax  .     .    (18) 

In  Eq.  (18)  ^  =  2.71828  is  the  base  of  the  Naperian  log- 
arithms ;  while  in  both  Eqs.  (16)  and  (18)  A,  Alt  A2  and  A3  are 
arbitrary  constants  to  be  determined  by  the  circumstances  of 
each  individual  case. 


CHAPTER   XII. 

WORKING  STRESSES  AND  SAFETY  FACTORS. 

Art.  83.— Definitions. 

IN  all  metallic  and  timber  constructions  the  greatest  (sup- 
posed) possible  loads  are  determined  from  the  attendant  cir- 
cumstances of  the  different  cases,  and  then  the  stresses  induced 
by  these  greatest  loads  are  computed.  These  stresses  are 
called  the  "  working  stresses." 

The  ultimate  resistance  of  any  piece  in  a  structure  divided 
by  the  working  stress  gives  a  number  called  the  "  safety  factor :" 
Occasionally  the  reciprocal  of  this  number  is  called  the  safety 
factor,  though  but  seldom. 

The  intensity  of  the  ultimate  resistance  of  any  piece  in  a 
structure  divided  by  the  intensity  of  the  working  stress,  will  also 
give  the  safety  factor.  This  is  the  more  usual  and  convenient 
form,  since  it  does  not  involve  the  cross  section  of  the  piece. 

The  values  of  safety  factors  depend  upon  many  circum- 
stances, such  as  kind  and  character  of  material,  kind  of  stress, 
circumstances  in  which  material  is  used  and  the  amount  of 
variation  of  stress  in  the  piece,  or  the  fatigue  of  the  material. 
The  safety  factor  is  intended,  also,  to  cover  both  computed 
stresses  and  others  which  are  recognized,  but  are  not  within  the 
reach  of  exact  analysis.  The  latest  practice  among  American 
engineers  will  be  illustrated  in  the  following  Articles  by  ex- 
tracts from  specifications  drawn  for  some  first-class  construc- 
tions. 


682  SPECIFICATIONS.'  [Art.  84. 

Art.  84.— Specifications  for  Sabula  Bridge. 

The  following  extracts  are  from  the  specifications  for  a 
bridge  at  Sabula,  on  the  line  of  the  Chicago,  Milwaukee  and 
St.  Paul  Railway. 

"  Quality  of  Iron  and  Steel. — All  eye  bars,  rods,  bolts,  and 
pins  shall  be  made  of  a  tough,  ductile,  fibrous  iron,  uniform  in 
quality,  and  which  shall  be  capable  of  withstanding  the  follow- 
ing tests  when  applied  to  full-sized  sections  of  the  material 
tested. 

Round  bars  up  to  \y2  inches  in  diameter  must  bend  double, 
or  until  inner  sides  are  in  contact,  when  cold,  without  showing 
signs  of  fracture. 

Square  bars  must  bend  cold  through  180  degrees  around  a 
cylinder  having  a  diameter  equal  to  two-thirds  the  length  of 
side,  without  showing  signs  of  fracture. 

Flats  must  bend  cold  through  180  degrees  around  a  cylinder 
having  a  diameter  equal  to  the  length  of  the  shortest  side, 
without  sign  of  fracture. 

The  ultimate  strength  of  the  bar  iron  used  shall  not  be  less 

/                   7.000  X  area  \ 
than  (52,000 — r-j-; }  pounds  per  square  inch  ;  area 

and  periphery  being  expressed  in  itaches. 

The  elastic  limit  shall  not  be  less  than  26,000  pounds  per 
square  inch,  and  the  elongation  of  the  bar  before  rupture  shall 
not  be  less  than  15  per  cent,  in  12  diameters. 

The  reduction  of  area  at  breaking  point  shall  not  be  less 
than  25  per  cent,  of  the' original  section. 

All  plate  and  shape  irons  used  in  tension  members,  or  in 
members  exposed  to  both  compressive  and  tensile  strains,  shall 
fulfill  all  the  foregoing  conditions  when  tested  in  specimens  of 
one  inch  area  and  15  inches  length  of  smallest  section,  except 
that  the  breaking  strain  per  square  inch  shall  not  be  less  than 
50,000  pounds  for  angles,  49,000  pounds  for  beams  and  channel 
iron,  and  48,000  pounds  for  plate  iron. 


Art.  84.]  SABULA   BRIDGE.  683 

Iron  (or  compression  members  must  be  tough,  fibrous,  uni- 
form in  quality,  and  with  an  elastic  limit  of  not  less  than  25,000 
pounds  per  square  inch. 

All  cast  iron  shall  be  good,  tough,  gray  iron  of  such  quality 
that  a  bar  five  feet  long,  one  inch  square,  and  4^  feet  between 
knife-edge  supports  will  sustain  a  weight  of  500  pounds  on 
knife  edge  at  middle  of  beam  before  breaking. 

All  steel  shall  be  of  a  good  quality  of  mild  steel,  having  an 
ultimate  tensile  strength  of  90,000  pounds,  or  over,  per  square 
inch,  an  elastic  limit  of  not  less  than  45,000  pounds,  a  ductility 
of  12  per  cent,  in  12  diameters,  and  not  less  than  15  per  cent, 
reduction  of  area  at  breaking  point. 

Specimens  one  square  inch  in  area  of  section  shall  bend 
cold  through  180  degrees  around  a  cylinder  whose  diameter  is 
4  times  the  length  of  the  shortest  side  of  the  test  piece. 

All  bar  and  rod  iron  shall  be  tested  in  full-sized  sections 
whenever  practicable. 

All  tension  iron  shall  be  rolled  from  piles  composed  of 
piling  pieces,  each  the  full  length  of  the  pile.  The  use  of  old 
rails  will  not  be  allowed  in  the  piles  for  this  grade  of  iron. 

Eye  Bars. —     ....... 

Bars  of  the  same  class  and  belonging  to  the  same  panel 
shall  be  drilled  at  the  same  temperature. 

No  error  in  length  of  bar  or  diameter  of  pin  hole  exceeding 
•g*¥  inch  will  be  allowed. 

The  section  of  metal  opposite  the  centre  of  pin  hole,  across 
the  eye,  shall  be  proportioned  according  to  the  following  table, 
the  diameter  of  the  bar  being  the  unit. 


684 


SPECIFICA  TIONS. 


[Art.  84. 


PIN. 

BAR. 

EYE  SECTION. 

Upset  heads  on  weldless  bar. 

Heads  rolled  on  bars. 

0.67 

I.O 

1-50 

1-33 

0-75 

x.o 

1.50 

1-33 

1.  00 

I.O 

1.50 

1.50 

1.25 

I.O 

I.  60 

1.50 

1-33 

I.O 

1.70 

1.  60 

1.50 

I.O 

I.85 

1.67 

i-75 

I.O 

2.00 

1.67 

2.00 

I.O 

2.  2O 

1-75 

Pins. — Pins  must  be  turned    . 

.     no  error  of  more  than  -fa  inch  in  diameter  being 
allowed. 

Pins  connecting  laterals  with  other  members  shall  be  turned 
down  to  a  diameter  of  not  more  than  T^-  inch  less  than  the  pin 
holes. 


Rods.—  ....... 

.  .  .  Screw  ends  shall  be  upset  so  as  to  give  10  per  cent, 
more  sectional  area  at  the  bottom  of  the  screw  thread  than  in 
the  body  of  the  bar.  .  .  . 

Chords  of  Pivot  Span. —         ..... 

.  .  .  No  error  exceeding  -fa  inch  in  length  of  part  or  in 
diameter  or  position  of  pin  hole  will  be  allowed.  The  pin  holes 
may  be  bored  -fa  inch  larger  than  the  pin,  but  this  is  the  utmost 


Art.  84.]  SABULA   BRIDGE.  685 

limit.     Rivet  holes  in  the  splices  and  in  all  steel  plates  shall 
be  punched   for  ^  inch  rivets  and  then  reamed   for  fa  inch 
rivets.  ....... 

Posts  of  Pivot  and  Fixed  Spans. —     . 


When  necessary,  pin  holes  in  posts,  chords  or  tie  struts 
shall  be  reinforced  by  additional  material,  which  must  contain 
rivets  enough  to  transmit  the  strain  to  the  original  member. 
The  open  sides  of  posts,  chords,  struts  and  tie  struts  shall  be 
connected  by  lattice  or  trellis  bars,  the  angles  of  which  shall 
not  exceed  63°  25'  for  single  bars  or  45°  for  double  bars  with 
riveted  intersections. 

The  unsupported  length  of  any  lattice  bar  shall  not  exceed 
50  times  its  thickness  ...... 

Riveted  Work. —  ..... 

Rivet  holes  shall  not  be  spaced  less  than  2j£  diameters 
between  centres,  nor  more  than  16  times  the  thickness  of  the 
thinnest  outside  plate  apart,  9  inches  being  the  maximum  pitch 
allowed  in  plate  riveting. 

No  rivet  holes  shall  be  less  than  i^  diameters  from  the  end 
of  a  plate,  or  i  y2  diameters  from  the  side  of  a  plate,  nor  ever 
less  than  I  j/£  inches  from  centre  of  hole  to  edge  of  plate,  ex- 
cept in  cases  where  the  plate  or  side  of  angle  is  less  than  2^/2 
inches. 

The  diameter  of  hole  shall  not  exceed  the  diameter  of  rivet 
more  than  -^  inch. 

When  two  or  more  thicknesses  of  plate  are  riveted  together, 
the  outer  row  of  rivets  shall,  if  practicable,  not  exceed  three 
rivet  diameters  from  the  edge  of  the  plate. 

Where  plates  more  than  12  inches  wide  are  used  in  the 
compression  flanges  of  girders  or  floor  beams,  an  extra  line  of 
rivets,  with  a  pitch  of  not  over  9  inches,  shall  be  driven  along 
each  edge  to  draw  the  plates  together. 


686  SPECIFICATIONS.  [Art.  85. 

Turn  Table. — This  will  be  rim-bearing, 

The  circular  girder  will  be  of  wrought  iron  26  feet  in  diam- 
eter, and  three  feet  in  depth,  and  will  be  made  of  the  grade  of 
iron  prescribed  for  material  in  tension.  The  wheels  will  be  18 
inches  in  diameter  and  8  inches  face,  and  will  be  turned  to  a 
uniform  size  and  bored.  (The  draw  or  pivot  span  is  360  feet 
long.) 

C.  SHALER  SMITH,  D.  J.  WHITTEMORE, 

Consulting  Engineer.  Chief  Engineer. 

Chicago,  Milwaukee  &  St.  Paul  Railway." 

These  specifications,  complete,  were  printed  in  the  "  Ameri- 
can Engineer,"  for  August,  1881. 


Art.  85. — Specifications  for  Albany  and   Greenbush   Bridge. 

These  specifications  were  prepared  by  Alfred  P.  Boiler,  C.  E., 
chief  engineer  of  the  Albany  and  Greenbush  Bridge  Co.  This 
bridge  crosses  the  Hudson  River  from  Albany  to  Greenbush, 
N.  Y.,  and  is  built  entirely  of  wrought  iron,  with  the  exceptions 
of  some  details  and  the  timber  of  the  flooring. 

The  following  are  the  portions  of  the  specifications  bearing 
upon  the  subject  under  consideration  : 

.     .     .     .     ;  the  maximum  stresses  shall  be  as  follows  : 

Tension. 

Floor  beam  hangers 6,500  Ibs.  per  sq.  inch. 

Counter  braces 8000  "  "  "  " 

Main  braces,  except  first  three  end  panels 10,000  "  "  "  " 

Lower  chord  and  upper  chord,  when  of  eye  bars,  also  first 

three  main  braces  at  ends 11,000  "  "  "  " 

Lower  chord  when  of  channel  bars,  taking  net  section 

only 10,000  "  "  "  " 

Wind  pressure  stresses 15,000  "  "  "  " 


Art.  85.]  GREE.VBUSff  BRIDGE.  68/ 


Compression  Members. 

Proportioned  according  to  Gordon's  formula,  with  the  fol- 
lowing constants  in  the  numerator  : 

For  Phoenix  columns n,ooo  Ibs.  per  sq.  inch. 

For  channel  and  beam  iron  sections 10,000    "      "     "       " 

Transverse   Stresses. 

On  pins 15,000  Ibs.  per  sq.  inch 

On  rolled  beams 10,000    "      "     "       " 

On  riveted  beams 8,000    "      "     "      " 

On  wooden  stringers 1,000    "      "     "       " 

Pressure  Stresses. 

On  bearing  surfaces  of  pins  (bearing  surface  =  diameter 

of  pin  x  width  of  bar) 12,500  Ibs.  per  sq.  inch. 

On  bearing  surface  of  rivets 10,000    "      "     "       " 

On  masonry 300    "      "     "       " 


Tests. — Full  sized  bars  of  flat,  round  or  square  iron,  having 
a  section  of  not  over  4.5  square  inches,  must  exhibit  an  ulti- 
mate resistance  of  50,000  pounds  per  square  inch,  and  stretch 
12.5  per  cent,  of  their  lengths.  Bars  of  larger  section  than  the 
above  will  be  allowed  a  reduction  of  1,000  pounds  per  square 
inch  for  each  additional  square  inch  of  sectional  area,  down  to 
a  minimum  of  46,000  pounds  per  square  inch.  Specimen  pieces 
taken  from  bars,  and  having  a  uniform  section  of  one  square 
inch  or  less  for  a  length  of  10  inches,  must  exhibit  the  follow- 
ing minimum  values  : 

From  bars  of  4.5  square  inches  in  section  and  under,  an 
ultimate  resistance  of  52,000  pounds  per  square  inch  with  a 
stretch  of  18  per  cent,  in  8  inches. 

From  bars  of  over  4.5  square  inches  in  section  a  reduction 
of  500  pounds  per  square  inch  for  each  additional  square  inch 


688  SPECIFICA  TIONS.  [Art.  86. 

of  section  down  to  a  minimum  of  50,000  pounds  per  square 
inch. 

Specimens  from  angles,  beams,  channels  or  plates  must 
show  an  ultimate  resistance  of  50,000  pounds  per  square  inch 
with  15  per  cent,  elongation  in  8  inches. 

All  iron  for  tension  members,  whether  bars,  angles  or  plates, 
must  permit  of  being  bent  cold,  without  cracking,  on  a  diam- 
eter not  greater  than  twice  the  thickness  of  the  bar,  plate  or 
angle  ;  the  cold-bend  test  on  angles  to  be  made  after  the  two 
legs  are  severed.  Any  of  the  above  classes  of  iron,  when 
nicked  and  broken,  must  exhibit  a  fibrous  fracture,  almost 
entirely  free  from  crystalline  specks. 

Turn  Table. —  . 

.  .  .  Centre  bearing  will  be  a  flat  pin,  proportioned  to 
a  pressure  of  6,000  pounds  per  square  inch,  with  two  steel 
(finally  made  of  phosphor  bronze)  discs,  the  whole  with  proper 
provision  for  oiling. 


Art.  86. — Niagara  Suspension   Bridge. 

In  his  "  Report  on  the  Renewal  of  the  Niagara  Suspension 
Bridge,"  Mr.  Leffert  L.  Buck,  C.  E.,  has  given  some  data  and 
calculations,  from  which  he  deduces  that  the  safety  factor  for 
the  cables  is : 

11,000  -v-  (1,400  X  1.78)  =  4.41. 

The  total  load  between  the  towers  being  1,400  tons  and 
the  ultimate  resistance  of  the  four  wrought-iron  cables,  11,000 
tons,  while  1.78  is  the  ratio  between  the  cable  tension  at  the 
top  of  the  towers  and  the  vertical  load  between  the  towers. 

The  new  iron  and  steel  stiffening  truss  is  designed  for  a 
safety  factor  of  5. 


Art.  S/.]  MENOMONEE  BRIDGE.  689 

In  the  towers,  built  of  limestone,  he  found  a  safety  factor 
of  nearly  23. 


Art.  87. — Menomonee   Draw-Bridge. 

The  specifications  for  this  wrought-iron  bridge  (located  on 
the  Chicago,  Milwaukee  and  St.  Paul  Railway)  were  written  by 
C.  Shaler  Smith,  C.  E.,  and  published  in  full  in  the  "Ameri- 
can Engineer,"  for  May,  1881. 

The  following  are  portions  of  these  specifications  : 


Quality  of  Iron. — The  iron  subject  to  tensile  strain  shall  be 
tough,  ductile,  and  of  uniform  quality,  capable  of  sustaining 
not  less  than  50,000  pounds  per  square  inch  of  sectional  area 
when  tested  in  large  and  long  lengths,  to  have  an  elastic  limit 
of  not  less  than  26,000  pounds  per  square  inch ; 


Tensile  Members. — The  tensile  members  shall  be  so  propor- 
tioned that  the  maximum  stresses  produced  by  the  weight 
of  the  structure,  and  the  specified  moving  and  engine  load 
shall  in  no  instance  exceed  the  following  : 


SQ. INCH. 

For  tensile  stresses  in  primary  members  or  those  upon  which  the  principal 

weight  comes  directly  from  the  floor  beam 8,000 

For  stresses  in  secondary  members,  or  those  which  receive  their  principal 

stresses  through  the  primaries 9,000 

Stresses  in  tertiary  members 10,000 

Stresses  in  end  suspenders 8,000 

Stresses  in  common  floor-beam  suspenders 4,000 

The  foregoing  are  the  stresses  to  be  used  where  the  speci- 
fied  members   are   eye  bars    or   bolts,  but   if  they   consist   of 
riveted  sections  the  stresses  allowed  shall  be  as  follows  : 
44 


690  SPECIFICATIONS.  [Art.  8/. 

LBS.  PER 
SQ. INCH. 

For  splice  plates  in  tension 7,000 

For  riveted  members  in  tension  in  chords 8,500 

For  riveted  members  in  tension  in  web 8,000 

When  any  member  is  exposed  to  stresses  in  opposite 
directions,  the  sections  shall  be  determined  by  the  following 
formula,  in  which  5  represents  the  sectional  area  in  square 
inches  required,  and  the  "  column  strength  per  square  inch  " 
of  sectional  area  is  determined  by  the  formulae  hereinafter 
given  : 

~  _     maximum  tension  maximum  compression 


10,000  column  strength  per  sq.  inch    . 

4 

Provided  that  the  section  thus  formed  shall  not  be  less 
than  required  by  the  foregoing  specifications  for  members  in 
tension,  or  the  following  specifications  for  members  in  com- 
pression. 

(For  these  formulae  see  Art.  50,  page  448.) 

Crippling  Stresses.  —  The  ultimate  crippling  resistance  in 
pounds  per  square  inch  of  section  of  the  several  forms  of  posts, 
struts  and  chords  will  be  determined  by  the  foregoing  for- 
mulae, in  which  /  -4-  d  or  H  equals  the  length  between  the  end 
bearings  in  terms  of  the  least  diameter. 

The  maximum  stress  permitted  in  any  purely  compressive 
member  will  be  the  quotient  resulting  from  dividing  the  ulti- 
mate resistance,  as  determined  by  the  above  formulae,  by  a 
coefficient  of  safety  (safety  factor)  equal  to  : 


"77,"  as  before,  being  the  measure  of  length  in  terms  of  least 
diameter. 


Art.  8/.J  MENOMONEE  BRIDGE.  69! 

Wind  Strains. —        ...... 

.  .  .  shall  be  resisted  by  lateral  and  vertical  rods  propor- 
tioned to  15,000  pounds  per  square  inch  in  tension,  and  lateral 
struts  proportioned  to  a  safety  factor  of  four  (4). 


Shearing  and  bending  stresses  at  the  lateral  connections 
.  .  .  shall  be  resisted  by  members  so  proportioned  that  the 
maximum  shearing  stresses  shall  not  exceed  10,000  pounds  per 
square  inch,  and  the  maximum  flexure  or  bending  stresses 
shall  not  exceed  22,500  pounds  per  square  inch. 

Floor  Beams  and  Track  Stringers. —  .  .  . 

.  .  .  the  resulting  stresses  (in  floor  beams)  .  .  .  shall 
not  exceed  8,000  pounds  per  square  inch  in  compression  or 
10,000  pounds  per  square  inch  in  tension. 

The  stringers  immediately  under  the  rails  shall  be  ... 
subject  to  the  same  conditions  as  to  limit  of  tensile  and  com- 
pressive  stresses  as  specified  above  for  floor  beams. 

If  the  floor  beams  are  of  built  sections,  the  rivets  must  be 
so  spaced  that  between  the  points  of  application  of  the  load 
and  the  points  of  support  there  are  rivets  enough  to  transmit 
the  flange  stresses  to  the  web  and  from  the  web  to  the  flange 
without  exceeding  a  shearing  stress  of  7,500  pounds  per  square 
inch  upon  the  rivets,  or  of  8,000  pounds  per  square  inch  of 
mean  pressure  on  the  semi-intrados  of  the  rivet  holes. 

Pins. — The  shearing  stress  on  any  pin  must  not  exceed 
7,500  pounds  per  square  inch  of  its  sectional  area.  The  stress 
per  square  inch  on  extreme  fibres,  caused  by  bending,  must 
not  exceed  15,000  pounds,  and  in  determining  this  bending 
stress  the  leverage  distance  shall  be  considered  as  from  centre 
of  eye  bar  to  centre  of  bearing,  or  of  opposite  eye  bar. 

No  pin  shall  have  a  less  diameter  than  two-thirds  of  the 
width  of  the  widest  bar  coming  upon  it. 

The  bearing  surface  of  any  pin  on  chord,  tie  or  post  shall 


692  SPECIFICATIONS.  [Art.  8/. 

not  be  exposed   to  a   greater  mean  compressive  stress  than 
8,000  pounds  per  square  inch. 

Riveted  Work. —         ...... 

All  rivets  with  crooked  heads  or  heads  not  formed  centrally 
on  the  shank,  or  rivets  which  are  loose  either  in  the  rivet  hole 
or  under  the  shoulder,  shall  be  cut  out  and  new  ones  put  in 
their  places. 

The  diameter  of  the  rivet  hole  shall  in  no  case  exceed  the 
diameter  of  the  rivet  by  more  than  T^  inch. 

In  members  consisting  of  two  or  more  pieces  of  shape  iron 
connected  by  lattice  or  lacing  bars,  there  shall  be  connection 
plates  at  each  end,  the  row  of  rivets  in  which  shall  be  not  less 
than  one  diameter  of  the  member  in  length.  ...  in  all 
riveted  work  the  distance  between  rivet  supports  across  the 
plate  shall  not  exceed  thirty  (30)  times  the  thickness  of  the 
plate,  and  no  closed  section  shall  have  members  of  less  thick- 
ness than  -f$  inch. 

All  rivets  in  splice  or  tension  joints  must  be  symmetrically 
arranged  so  that  each  half  of  a  tension  member  or  splice  plate 
will  have  the  same  uncut  area  on  each  side  of  its  centre  line. 

No  rivet  shall  be  exposed  to  more  than  7,500  pounds  per 
square  inch  in  shear,  or  more  than  8,000  pounds  mean  pressure 
per  square  inch  of  semi-intrados. 

Bed  Plates  and  End  Supports. — The  bed  plate  supporting 
turn-table  centre  shall  be  so  proportioned  that  the  pressure  on 
the  masonry  shall  not  exceed  25,000  pounds  per  square  foot 
while  the  span  is  rotating,  and  the  pressure  of  the  wheel  track 
and  end  supports  of  span  on  the  masonry  shall  not  exceed  this 
limit  when  the  bridge  is  fully  loaded  as  heretofore  specified. 

Turn  Table. — 


Art.  87.]  MENOMONEE  BRIDGE.  693 

If  a  wearing  or  friction  centre  pin  is  used,  the  pressure 
while  rotating  shall  not  exceed  6,000  pounds  per  square  inch. 

If  a  lubricated  centre  pin  is  used,  the  weight  while  turning 
must  not  exceed  1,000  pounds  per  square  inch  on  the  spindle 
bearing. 

If  a  Sellers'  centre  is  used,  the  rotating  load  per  lineal  inch 
on  the  steel  rollers  shall  not  exceed  that  given  by  the  follow- 
ing formula  for  steel  rollers  on  steel  plates  : 


P  =  1/3,072,000^; 

in  which  formula  P=  pressure  per  lineal  inch  of  roller,  and  d 
the  mean  diameter  in  inches. 

The  load  per  lineal  inch  of  face  of  wheel  (for  rim-bearing 
table),  while  the  span  is  turning,  shall  not  exceed  that  given  by 
the  formula  : 


P  = 

for  cast-iron  wheels  upon  wrought-iron  wheel  track. 

For  cast-iron  wheels  upon  cast-iron  track,  the  load  per 
lineal  inch  of  face  of  wheel,  while  the  span  is  turning,  shall  not 
exceed  that  given  by  the  formula  : 


P  =   A/222, 

For  steel  wheels  upon  steel  track,  use  the  following  formula 
as  to  limit  of  pressure  upon  lineal  inch  of  wheel  face  : 


p  —  V  i,  296,000^. 

For  steel  wheels  upon  wrought-iron  track  : 


p  =  V  i, 024,000^. 


694  SPECIFICATIONS.  [Art.  88. 

And  for  steel  wheels  upon  cast-iron  wheel  track,  use  : 
P=  V i, 6 1 7,000^  . 


Workmanship,  Painting,  etc. —  .... 

The  parts  composing  the  posts  or  struts  must  be  of  entire 
lengths,  without  splicing  between  end  bearings. 

Tests.—  ....... 

All  bars  subject  to  tensile  stress  shall  be  tested  by  the  con- 
tractor, .  .  .  ,  to  18,000  pounds  per  square  inch  of  sec- 
tional area." 

Art.  88.— Franklin  Square   Bridge. 

The  specifications  from  which  the  following  portions  are 
selected,  apply  to  the  construction  of  a  bridge  over  Franklin 
Square,  New  York  City,  on  the  line  of  the  East  River  Bridge  ; 
they  were  prepared  by  the  Chief  Engineer,  Washington  A. 
Roebling,  C.E. 

10.  All  members  shall  be  so  designed  that  the  stresses 
coming  upon  them  can  be  accurately  calculated. 

12.  For  wrought  iron  the  following  clauses  apply  : 

I.  Members  in  tension    shall  be  so  proportioned  that  the 

maximum   stresses  shall  not  cause    greater  tensions  than  the 

following  : 

LBS.    PER  SQ.    IN. 

On  lateral  bracing 15,000.00 

On  solid  rolled  beams  used  as  cross  floor  beams  and  stringers. .  10,000.00 

On  bottom  chords  and  main  diagonals 10,000.00 

On  counter  rods  and  long  verticals 8,000.00 

On  bottom  flanges  of  riveted  cross  girders,  net  section 8,000.00 

On  members  liable  to  sudden  loading  and  shocks 6,000.00 


Art.  88.]  FRANKLIN  SQUARE  BRIDGE.  695 

II.  Members  in  compression  shall  be  so  proportioned  that 
the  maximum  load  shall  not  cause  a  greater  compression  than 
that  determined  by  the  formulae  : 

8,000  , 

Jr  —  -  — j^ ,  for  square  ends. 

i  + 


8,000  , 

P  =  -  — -fr ,  for  one  square  end  and  one  pin  end. 


3o,ooo^2 


8,000  .        .    , 

P  —  -          —j-2  -  ,  for  pin  bearing  at  each  end. 


20,OOOR* 


P  —  allowed  compression  per  sq.  in.  of  cross  section. 
/   =  length  of  compression  member  in  inches. 
R  =  least  radius  of  gyration  in  inches. 

III.  The  lateral  struts  shall  be  proportioned  by  the  above 
formulae  to  resist  the  resultant  due  to  an  initial  stress  pro- 
duced by  adjusting  the  bridge,  assumed  at  10,000  pounds  per 
sq.  in.  u^)on  all  rods  attached  to  struts. 

IV.  In  beams  and  girders,  compression  shall  be  limited  as 
follows  : 

LBS.    PER  SQ.   IN. 

In  rolled  floor  beams  used  as  cross  floor  beams  and  stringers.  .  .    10,000.00 
In  riveted  plate  girders  used  as  cross  floor  beams  .............     6,000.00 

In  any  riveted  girder  under  20  feet  long  ..........  ...........     5,000.00 

V.  Members  subject  to  alternate  stresses  of  tension  and 
compression  shall  be  proportioned  to  resist  each  of  them,  but 
the  sectional  area  of  such  members  shall  be  increased  if  the 
engineer  requires  it. 


696  SPECIFICATIONS.  [Art.  88. 

VI.  The  diameter  of  a  pin  shall  not  be  less  than  two-thirds 
of  the  largest  dimension  of  any  member  attached  to  it,  and  its 
effective  length  shall  not  be  greater  than  four  times  its  diam- 
eter, plus  the  breadth  of  the  foot  of  the  post  through  which  it 
passes.     The  shearing  stress  upon   any  pin   shall  not  exceed 
7,500  pounds  per  sq.  in.  ;  the  crushing  stress    upon  the  pro- 
jected area  of  the  semi-intrados  (diameter  multiplied  by  thick- 
ness of  the  piece)  of  any  member  connected  to  the  pin,  shall 
not  exceed  15,000  pounds  per  sq.  in.,  nor  when  the  centres  of 
the  bearings  of  the  strained  members  are  taken  as  the  points 
of  application  of  the  stresses,  shall  the  bending  stress  exceed 
15,000  pounds  per  sq.  in. 

VII.  Plate  girders  shall  be  proportioned  upon  the  supposi- 
tion that  the  bending  or  chord  stresses  are  resisted  entirely  by 
the  upper  and  lower  flanges,  and  that  the  shearing  or  web 
stresses  are  resisted  entirely  by  the  web  plates. 

VIII.  When  the  length  of  beams  and  girders  is  more  than 
thirty  times  their  width,  their  flanges  in  compression  shall  be 
stayed  against  transverse  crippling,  and  the  unsupported  width 
of  any  plate  in  compression  shall  not  exceed  thirty  times  its 
thickness. 

IX.  Shearing  stresses  in  web  plates  shall  not  be  greater 
than  4,000  pounds  per  sq.  in.,  and  no  web  or  similar  plate  shall 
be  less  than  T8T  inch  in  thickness.     When  the  least  thitkness  of 
the  web  is  less  than  ¥V  the  depth  of  a  girder,  the  web  shall  be 
stiffened  at  intervals  not  over  twice  the  depth  of  the  girder. 

X.  All   rivets   and    bolts   connecting   parts    of   girders   or 
trusses  must  be  so  spaced  that  the  shearing  stress  shall  not 
exceed  7,500  pounds  per  sq.  in. 

XL  In  members  subject  to  tensile  stress,  full  allowance  shall 
be  made  for  reduction  of  section  by  rivet  holes  or  otherwise. 

XII.  Any  member  subjected  to  a  bending  stress  from  local 
loads,  in  addition  to  the  stress  produced  by  its  position  as  a 
member  of  the  structure,  must  be  proportioned  to  resist  the 
combined  stresses. 


Art.  88.]  FRANKLIN  SQUARE  BRIDGE.  697 

13.  The  stresses  allowed  in  members  made  of  steel,  gener- 
ally, will  be  one-half  larger  than  those  specified  for  wrought 
iron  ;  .  .  . 

Eye-Bars  and  Pins. 
•  ••••••• 

1 6.  Bars  which  are  to  be  placed  side  by  side  in  the  struct- 
ure shall  be  bored  at  the  same  temperature,  and  of  such  equal 
lengths  that  when  the  bars  are  piled  on  each  other  the  pins 
may  pass  through  all  the  holes  at  both  ends  without  driving. 

17.  Pin  holes  shall  not  be  bored  more  than  -f0  inch  larger 
than  the  diameter  of  the  corresponding  pins.     .     . 

1 8.  Any  loop  attached  to  a  pin  must  fit  it  perfectly  through- 
out its  semi-circumference. 


Compression  Members — Riveted  Work. 

23.  The  open  sides  of  all  trough-shaped  sections  shall,  at 
distances  not  exceeding  the  width  of  the  member,  be  stayed 
diagonally  by  lattice  bars  in  size  duly  proportioned  to  such 
width. 

24.  Whenever  it  is  necessary  to  reduce  the  pressure  upon  a 
pin  to  the  limit  prescribed,  the  pin  holes  shall  be  reinforced  by 
additional  plates,  which  must  contain  enough  rivets  to  transfer 
the  proportion  of  pressure  to  the  member. 

27.  The  pitch  of  rivets  in  all  classes  of  work  shall  not  ex- 
ceed six  inches,  nor  sixteen  times  the  thickness  of  the  thinnest 
outside  plate,  nor  be  less  than  three  diameters  of  the  rivet ; 
the  rivets  used  shall  be  generally  ^  and  7/&  inch  in  diameter. 
In  compression  members  the  pitch  of  rivets  for  a  space  from 
its  end  of  twice  the  breadth  or  width  of  a  member,  shall  not 
be  over  four  times  the  diameter  of  the  rivets. 


698  SPECIFICATIONS.  [Art-  88. 

Bed  and  Buckled  Plates. 

38.  All  bed  plates  shall  be  of  such  dimensions  that  the 
greatest  pressure  upon  the  masonry  shall  not  exceed  250 
pounds  per  sq.  in. 

40.  The  buckled  plates  for  the  roadways  shall  not  be  less 
than  inch  thick. 


Tests  of  Material. 

47.  All  wrought  iron  must  be  tough,  fibrous  and  uniform  in 
character,  and  shall  have  a  limit  of  elasticity  of  not  less  than 
26,000  pounds  per  sq.  in.  ..... 

52.  The  steel  used  must  be  of  a  uniform  and  suitable  qual- 
ity, known  as  mild  steel.  It  must  have  an  ultimate  tensile 
strength  of  70,000  pounds  per  sq.  in.  of  full  section,  and  when 
marked  off  in  foot  lengths,  have  an  ultimate  stretch  of  10  per 
cent,  in  the  total  length,  and  of  at  least  15  per  cent,  in  the  foot 
which  includes  the  fractured  section,  showing  a  reduction  in 
section  at  the  point  of  fracture  of  at  least  25  per  cent.  It  must 
have  an  elastic  limit  of  40,000  pounds  per  sq.  in.,  and  a  modu- 
lus of  elasticity  between  26,000,000  and  30,000,000  pounds. 
Small  specimens,  one  foot  long  and  of  uniform  section  of  one 
sq.  in.,  cut  without  work  from  finished  shapes,  shall  have  an 
ultimate  resistance  of  75,000  pounds  and  stretch  15  per  cent. 

55.  Castings  must  be  smooth,  free  from  air  holes,  cinders, 
and  other  imperfections,  and  of  good,  tough  cast  iron  ;  gener- 
ally they  shall  not  be  less  than  ^  inch  thick. 


Art.  90.]  RAIL  WA  Y  SPECIFICA  TIONS.  699 


Art.  89. — General  Specifications. 

Some  general  specifications  in  use  at  the  present  time  by 
one  of  the  large  railroads  of  the  country,  require  that  the 
working  stresses  per  square  inch  shall  not  exceed  the  follow- 
ing values  : 

LBS. 

Wrought  iron  in  tension 10,000.00 

Wrought  iron  in  compression 8,000.00 

Wrought  iron  rivets  in  shear 6,500.00 

Shearing  in  web  plate  of  plate  girders 5,000.00 

Shearing  in  pins   • 6,000.00 

Timber  in  tension 400.00  to  500.00 

Timber  in  compression 200.00  to  300.00 


Art.  90. — New  York,  Chicago  &  St.  Louis  Railway  Specifications. 

"  Tensile  Members. — 

Where  the  floor  beams  are  suspended  the  loops  shall  be 
double  at  each  panel  joint,  and  the  strain  .  .  .  must  not 
exceed  4,000.00  pounds  per  sq.  in 

Compression  Members — . 

The  thickness  of  metal  in  columns  must  not  be  less  than 
-gV  of  the  width  of  plates  between  supports,  nor  less  than  ^ 
inch  when  both  faces  are  accessible  for  painting,  and  -f^  inch 
when  one  face  only  is  accessible.  ... 

When  lattice  work  is  used,  the  distance  between  rivets 
must  not  be  less  than  length  of  segment  of  equal  strength  per 
sq.  in.,  as  the  column  itself  and  the  lattice  bars  must  be  calcu- 
lated as  struts  resisting  the  difference  in  the  strengths  per  sq. 
in.  of  the  column,  and  that  of  its  weakest  segment  acting 
singly  without  lateral  supports. 

In  I-beams  the  compression  per  sq.  in.  in  the  compressed 
flanges  must  not  exceed 


;oo 


SPECIFIC  A  TIONS.  [Art.  90. 


I         4O,OOO 


where  /  =  length  of  beam  in  inches  and  b  —  breadth  in  inches 
of  the  compressed  flange.  The  shearing  stress  per  sq.  in.  in 
web  of  I-beam  must  not  exceed 

I        40,000 

$:    ~^~; 


1    3,ooo/2 

in  which  d  =  distance  in  inches  between  flanges  or  stiffeners, 
measured  on  a  line  inclined  at  45°,  and  /  =  thickness  of  web 
in  inches. 

Connections  and  Attachments. —      ..... 

Tensile  stresses  will  not  be  allowed  in  a  transverse  direction 
to  the  fibres  of  the  iron.  Shearing  stresses  will  not  be  allowed 
in  a  direction  parallel  to  the  fibres  of  the  iron. 


Pins  and  Rivets. —     .  .  .  . 

.     .     .     the  mean  pressure  on  semi-intrados  of  pin  and  rivet 
holes  must  not  be  more  than  12,500.00  pounds  per  sq.  in.     .     . 


For  Bending,  the  maximum  allowed  on  the  outside  fibres  of 
timber  shall  be  1,000.00  pounds  per  sq.  in. 

Bed  Plates  and  Friction  Rollers.       f 

.  .  .  the  friction  rollers  must  be  so  proportioned  that  the 
pressure  per  lineal  inch  of  roller  does  not  exceed  -y7 5 40,000  X  d% 
in  which  d  represents  the  diameter  of  roller  in  inches." 


Art.  QI.]  PLATTSMOUTH  BRIDGE.  70 1 


Art.  91. — Plattsmouth  Bridge. 

The  following  information  in  regard  to  the  combined  steel 
and  iron  bridge  at  Plattsmouth,  Neb.  (George  S.  Morrison, 
Chief  Engineer),  is  taken  from  the  "Railroad  Gazette"  of  I7th 
Dec.,  1880.  This  bridge  carries  a  single  track  for  the  C.  B.  & 
Q.  R.R. 

"  The  top  chords  and  inclined  end  posts  are  riveted  steel 
members  formed  of  plates  and  angles,  measuring  28  ins.  wide 
by  19  ins.  deep  over  all,  the  under  side  being  open  and  laced. 
In  the  manufacture  of  these  pieces  the  steel  was  first  punched 
with  ^-inch  holes,  then  assembled  and  the  holes  reamed  to  I 
inch,  and  then  riveted  without  taking  apart,  the  rivets  being  of 
low  carbon  steel.  The  maximum  compressive  strain  allowed 
to  these  members  is  15,000.00  pounds  per  sq.  in.,  the  sections 
being  so  proportioned  as  to  carry  this  strain  on  the  two  side 
pieces  of  the  member,  the  central  part  of  the  top  plate  being 
relied  upon  only  for  lateral  stiffness.  The  connection  between 
the  top  chord  and  end  posts,  and  between  the  end  posts  and 
bolsters  are  pin  connections,  all  parts  being  entirely  of  steel. 
On  these  pins  the  pressure  per  square  inch,  measured  on  the 
diameter  and  not  on  the  semi-intrados,  is  limited  to  20,000 
pounds  per  sq.  in. 

The  steel  bars  in  the  bottom  chord  and  the  main  ties  were 
rolled  by  the  Kloman  process  in  a  universal  mill,  the  motion 
being  reversed  while  the  bar  is  still  between  the  rolls,  the  heads 
being  subsequently  forged  into  shape  with  a  steam  hammer, 
and  the  whole  bar  afterwards  annealed.  Of  seven  full-sized 
bars  which  were  tested  to  breaking,  not  one  broke  or  showed 
any  weakness  in  or  near  the  head.  .  .  The  maximum  strain 
allowed  on  steel  in  tension  is  15,000  pounds  per  sq.  in.,  this 
occurring  only  in  the  middle  panels  of  the  bottom  chord,  and 
being  reduced  to  12,500  in  the  end  panels;  in  the  web  the 
strain  per  square  inch  varies  from  10,000  pounds  at  the  centre 


7O2  SPECIFICATIONS.  [Art.  91. 

to  about  12,500  in  the  end  ties,  except  under  the  extraordinary 
supposition  of  the  entire  weight  on  the  driving  wheels  of  two 
75-ton  locomotives  being  carried  entirely  by  the  same  system  ; 
in  this  case  the  maximum  strain  on  the  end  diagonals  will 
slightly  exceed  14,000  pounds  per  sq.  in. 

The  counter  ties  and  lateral  rods  are  also  of  steel.  .  .  The 
strain  on  the  counters  is  always  less  than  10,000  pounds  per 
sq.  in.  ;  that  on  the  laterals  is  limited  to  22,000  pounds.  Tests 
made  of  these  light  steel  bars  showed  a  superior  proportional 
excellence  fully  equal  to  that  commonly  found  in  small  sec- 
tions of  wrought  iron  as  compared  with  large  sections. 

The  intermediate  posts  are  of  wrought  iron,  each  post  con- 
sisting of  two  channels  laced  at  the  sides. 

All  the  pins,  except  the  lateral  ones,  are  of  4|f  inches  diam- 
eter. 

Steel. 

The  steel  used  in  the  Plattsmouth  bridge  was  manufactured 
by  Hussey,  Howe  &  Co.,  of  Pittsburgh,  in  an  open  hearth  fur- 
nace. The  specifications  required  that  a  sample  (about  $/Q  inch 
diameter)  should  be  taken  from  every  melt,  and  that  this  bar 
should  bend  cold  180  degrees  around  its  own  diameter  without 
cracking ;  that  it  should  have  an  elastic  limit  of  at  least  50,000 
pounds,  and  an  ultimate  strength  of  at  least  80,000  pounds ; 
and  that  it  should  elongate  12  per  cent,  before  breaking  and 
show  a  reduction  of  20  per  cent,  at  the  point  of  fracture.  The 
percentage  of  carbon  was  fixed  at  0.35. 

A  difference  in  the  strength  of  large  and  small  sized  bars 
corresponding  to  that  which  exists  in  iron  bars  was  found  in 
the  steel.  The  finished  bars  measured  6  x  ij£  inches  to  \% 
inches;  when  tested  in  the  Government  machine  at  Water- 
town,  Mass.,  they  were  found  to  have  an  elastic  limit  of  37,000 
pounds,  and  ultimate  resistances  of  66,000  to  73,000  pounds 
per  square  inch.  The  modulus  of  elasticity  below  the  elastic 


Art.  92.]  STEEL   CABLE    WIRE.  703 

limit  was  exceedingly  uniform.  Smaller  sizes,  used  in  counters 
and  laterals,  approximated  closely  in  their  strength  and  elastic 
limit  to  the  test  samples. 


Art.  92.— Specifications  for  Steel  Cable  Wire  for  the  East  River 
Suspension  Bridge. 


3.  The  general  character  of  the  wire  is  as  follows  :  it  must 
be  made  of  steel ;  it  must  be  hardened  and  tempered  ;  and, 
lastly,  it  must  all  be  galvanized. 

4.  The  size  of  the  wire  shall  be  No.  8  full,  Birmingham 
gauge.  .  .  .  . 

5.  Each  wire  must  have  a  breaking  strength  of  no  less  than 
3,400  pounds.     This  corresponds  in  wire  weighing  14  feet  to 
the  pound,  to  a  rate  of  160,000  pounds  per  square  inch  of  solid 

section.     The  elastic  limit  must  be  no  less  than  -~-  of  the 

100 

breaking  strength,  or,  1,600  pounds.  Within  this  limit  of  elas- 
ticity, it  must  stretch  at  a  uniform  rate  corresponding  to  a 
modulus  of  elasticity  of  not  less  than  27,000,000  nor  exceed 
29,000,000.  ....... 

Mode  of  Testing. 

There  will  be  four  kinds  of  tests. 

Firstly. — One  ring  in  every  forty  (40)  will  be  tested  as  fol- 
lows :  a  piece  of  wire  sixty  (60)  feet  long,  will  be  cut  off  from 
either  end  of  the  ring,  and  it  will  then  be  placed  in  a  vertical 
testing  machine.  An  initial  strain  of  400  pounds  is  now  ap- 
plied, which  should  take  out  every  crook  and  bend.  A  vernier 

gauge,  capable  of  being  read  to  -         -  of  one  foot,  is  so  at- 


SPECIFICATIONS.  [Art.  92. 


tached  as  to  indicate  the  stretch  of  50  feet  of  the  wire.  Suc- 
cessive increments  of  400  pounds  strain  are  then  applied,  and 
the  vernier  read  each  time,  until  a  strain  of  1,600  pounds  is 
reached. 

The  conditions  now  are  as  follows  :  that  the  amount  of 
stretch  for  each  of  these  increments  shall  be  the  same,  and  that 
the  total  stretch  between  the  initial  and  terminal  strains  shall 

not  be  less  than    ^       of  one  foot,  equal  to  —  —  —  •  of  the  50 
1,000  100,000 

feet.  And  furthermore,  on  reducing  the  strain  to  1,200 
pounds  there  shall  be  a  permanent  elongation  not  exceeding 

-  of  its  length. 
100,000 

The  same  wire  will  then  be  subjected  to  a  breaking  strain, 
and  the  total  amount  of  stretch  noted.  The  minimum  strength 
required  is  3,400  pounds,  equal  to  an  ultimate  strength  of 
160,000  pounds  per  square  inch.  The  minimum  stretch,  when 
broken,  shall  have  been  2  per  cent,  in  50  feet,  and  the  diameter 

of  the  wire  at  the  point  of  fracture  shall  not  exceed  —  —  of  one 

100 

inch. 


Fourthly. — Every  ring  will  be  subjected  to  a  bending  test 
by  cutting  off  from  each  ring  a  piece  of  wire  one  foot  long,  and 
coiling  it  closely  and  continuously  around  a  rod  one  half  inch 
in  diameter,  when,  if  it  breaks  it  will  be  rejected. 


Straight  Wire. 

9.  All  the  wire  .  .  .  must  be  "  straight  "  wire  ;  that  is 
to  say,  when  a  ring  is  unrolled  upon  the  floor  the  wire  behind 
must  lie  perfectly  straight  and  neutral,  without  any  tendency 
to  spring  back  in  the  coiled  form,  as  is  usually  the  case.  This 


Art.  94.]  STEEL    WIRE  ROPES.  705 

straight  condition  must  not  be  produced  by  the  use  of  straight- 
ening machines  of  any  kind,  as  they  only  injure  the  strength 
and  elasticity  of  the  wire.  .  .  .  .  ." 


Art.  93. — Specifications  for  Steel  Wire  Ropes  for  the  Over-Floor  Stays 
and  Storm  Cables  of  the  East  River  Suspension  Bridge. 


3.  The  steel  from  which  the  wire  for  these  ropes  is  made 
must  be  of  a  uniform   and  suitable  quality,  and  after  drawn 
must  be  thoroughly  and  evenly  galvanized  throughout. 

4.  The  galvanized  wire  must  have  an  ultimate  strength  of 
150,000  pounds  per  square  inch  of  full  section.     When  tested 
in  lengths  of  five  feet  it  must  stretch  no  less  than  three  and 
one-half  per  cent,  of  its  length,  and  in  lengths  of  one  foot  it 
must  stretch  no  fess  than  four  per  cent. 

5.  It  must  be  capable  of  being  bent  continuously  around  a 
rod  of  three  times  the  diameter  of  the  wire,  without  fracture. 

6.  The   modulus  of  elasticity   must   not    vary   more   than 
2,000,000  pounds,  nor  exceed  30,000,000  pounds. 

7.  It  must  have  a  limit  of  elasticity  of  not  less  than  70,000 
pounds  per  square  inch. 


Art.  94. — Specifications  for  Steel  Suspenders,  Connecting  Rods,  Stirrups 
and  Pins,  for  the  East  River  Suspension  Bridge. 


All  of  the  steel  used  must  be  of  a  uniform  and  suitable 
quality,  known  as  "  Mild  Steel."  It  must  have  an  ultimate 
tensile  strength  of  75,000  pounds  per  square  inch  of  full  sec- 
tion, and  an  ultimate  stretch  of  no  less  than  15  per  cent,  in  one 
foot  of  length,  including  the  fractured  section  ;  and  a  reduc- 
tion of  no  less  than  25  per  cent,  of  area  at  the  point  of  fracture. 
It  must  have  an  elastic  limit  of  no  less  than  45,000  pounds  per 
square  inch,  and  a  modulus  of  elasticity  between  26,000,000 
45 


706  SPECIFICATIONS.  [Art.  95. 

and  30,000,00x3  pounds  per  square  inch.  Specimens  turned 
down  from  full-sized  rods  to  an  area  of  one  square  inch,  or  less, 
must  show  a  greater  strength  per  square  inch,  and  a  greater 
elongation  than  that  called  for  in  the  full  section." 


Art.  95.— Specifications  for  Certain  Steel  Work    .    .    .     East  River 

Bridge,  1881. 


'  All  of  the  steel  used  in  this  work  must  be  of  a  mild,  uni- 
form, elastic  and  ductile  quality,  suitable  for  bridge  members. 
Siemens-Martin  or  open-hearth  steel,  or  Bessemer  steel  under 
the  Hay  process,  will  be  preferred. 

Specimens  of  the  steel  proposed  to  be  used  must  be  fur- 
nished by  each  bidder.  Two  specimens,  direct  from  the  rolls, 
each  I  inch  square  and  24  inches  long,  are  required. 

All  of  the  steel  must  be  capable  of  sustaining  a  tensile 
strain  in  every  full-sized  round  or  flat  bar  of  not  less  than 
70,000,000  pounds  per  square  inch  of  cross  section.  It  must 
have  an  elastic  limit  in  all  shapes  of  no  less  than  40,000  pounds 
per  square  inch.  A  modulus  of  elasticity  of  not  less  than 
26,000,000  nor  more  than  30,000,000  pounds  per  square  inch. 

An  ultimate  elongation  of  10  per  cent,  of  the  full  length  of 
uniform  sections,  and  15  per  cent,  in  one  foot  of  length,  inclu- 
sive of  fractured  section,  is  also  required.  The  area  of  the 
reduced  section  at  the  point  of  fracture  must  not  exceed  80 
per  cent,  of  the  original  section. 

Small  specimens  of  one  foot  in  length,  of  even  section  of 
one  square  inch,  or  less,  should  reach  in  tensile  strength  75,000 
pounds  per  square  inch,  with  a  modulus  and  limit  of  elasticity, 
and  reduction  of  area  before  mentioned,  and  an  ultimate 
stretch  of  15  per  cent. 

All  round  or  flat  bars,  or  flat  pieces  cut  from  the  web  of  any 
shaped  bars,  must  be  capable  of  being  bent  cold  for  1 80°  to  a 


Art.  95.]  EAST  RIVER  STEEL    WORK.  707 

curve  whose  diameter  is  no  greater  than  the  thickness  of  the 
bar,  and  that  without  cracking. 

The  rivets  must  be  made  of  very  ductile  steel  particularly 
adapted  for  that  use. 

The  rods  from  which  the  rivets  are  made  must,  when  tested, 
have  a  tensile  strength  of  not  less  than  70,000  pounds  per 
square  inch,  and  elongate  at  least  20  per  cent,  in  a  length  of 
one  foot,  and  shall  reduce  at  the  point  of  fracture  30  per  cent. 

If  the  minimum  is  reached  in  any  one  of  these  requirements 
the  others  must  be  exceeded  by  at  least *io  per  cent.  The  rod 
must  be  capable  of  being  bent  cold  under  a  hammer  180°,  and 
the  inner  surfaces  brought  in  contact  without  producing  any 
fracture. 


Cold  straightening  must  be  avoided,  and  when  resorted  to, 
the  piece  so  straightened  must  be  annealed  afterwards,  and  all 
pieces,  of  which  any  portion  for  any  cause  is  reheated,  the 
whole  must  be  annealed  and  very  slowly  cooled  ;  and  all  pieces 
in  which,  from  test  or  otherwise,  a  want  of  uniformity  is  sus- 
pected, must  be  annealed  if  required  by  the  engineer. 

All  rivet  holes  must  be  drilled,  unless  some  system  of 
punching  and  reaming  approved  by  the  engineer  be  followed, 
whereby  all  of  the  compressed  section  around  the  punched 
hole  be  cut  away. 

The  spacing  must  be  accurately  done,  as  no  gauging  or 
drifting  will  be  allowed." 


CHAPTER    XIII. 

THE  FATIGUE  OF  METALS. 

Art.  96.— Woehler's  Law. 

IN  all  the  preceding  pages,  that  force  or  stress,  which,  by  a 
single  or  gradual  application,  will  cause  the  failure  or  rupture 
of  a  piece  of  material,  has  been  called  its  "  ultimate  resistance." 
It  has  long  been  known,  however,  that  a  stress  less  than  the 
ultimate  resistance  may  cause  rupture  if  its  application  be  re- 
peated (without  shock)  a  sufficient  number  of  times.  Preced- 
ing 1859  no  experiments  had  been  made  for  the  purpose  of 
establishing  any  law  connecting  the  number  of  applications 
with  the  stress  requisite  for  rupture,  or,  with  the  variation 
between  the  greatest  and  least  values  of  the  applied  stress. 

During  the  interval  between  1859  and  1870,  A.  Wohler, 
under  the  auspices  of  the  Prussian  Government,  undertook  the 
execution  of  some  experiments,  at  the  completion  of  which  he 
had  established  the  following  law  : 

Rupture  may  be  caused  not  only  by  a  force  which  exceeds  the 
ultimate  resistance,  but  by  the  repeated  action  of  forces  alternately 
rising  and  falling  between  certain  limits,  the  greater  of  which  is 
less  than  the  ultimate  resistance ;  the  number  of  repetitions  re- 
quisite for  rupture  being  an  inverse  function  both  of  this  vari- 
ation of  the  applied  force  and  its  upper  limit. 

This  phenomenon  of  the  decrease  in  value  of  the  breaking 
load  with  an  increase  of  repetitions,  is  known  as  "  the  fatigue 
of  materials'^ 

Although  the  experimental  work  requisite  to  give  Wohler' s 


Art.  97.]  EXPERIMENTAL  RESULTS.  709 

law  complete  quantitative  expression  in  the  various  conditions 
of  engineering  constructions  can  scarcely  be  considered  more 
than  begun,  yet  enough  has  been  done  by  Wohler  and  Span- 
genberg  to  establish  the  fact  of  metallic  fatigue,  and  a  few 
simple  formulae,  provisional  though  they  may  be.  The  im- 
portance of  the  subject  in  its  relation  to  the  durability  of  all 
iron  and  steel  structures  is  of  such  a  high  character  that  a 
synopsis  of  some  of  the  experimental  results  of  Wohler  and 
Spangenberg  will  be  given  in  the  next  Article. 

Art.  97. — Experimental    Results. 

The  experiments  of  Wohler  are  given  in  "  Zeitschrift  fur 
Bauwesen,"  Vols.  X.,  XIII.,  XVI.  and  XX.,  and  those  of  Span- 
genberg may  be  consulted  in  "  Fatigue  of  Metals,"  translated 
from  the  German  of  Prof.  Ludwig  Spangenberg,  1876. 

These  results  show  in  a  very  marked  manner  the  effect  of 
repeated  vibrations  on  the  intensity  of  stress  required  to  pro- 
duce rupture. 

Spangenberg  states  that  "  the  experiments  show  that  vibra- 
tions may  take  place  between  the  following  limits  with  equal 
security  against  rupture  by  tearing  or  crushing : 

f -f  17,600  and  —  17,600  Ibs.  per  sq.  in. 
Wrought  iron \  -{-  33,000  and  o 

I  -j-  48,400  and  -f-  26,400 

I  -f-  30,800  and  —  30,800 
Axle  cast  steel <  -j-  52,800  and  o 

[-{-  88,000  and  -f  38,500 

f-f-  55,000  and  o 

Spring  steel  not  hardened..  J  +  77,°°°  and  +  27,500 
|  -|-  88,000  and  +  44,000 
[-{-  99,000  and  -f-  66,000 

And  for  axle  cast  steel  in  shearing : 

-\-  24,200  and  —  24,200  Ibs.  per  sq.  in. 
-f  41,800  and  o       "      "    "    " 


710 


FATIGUE  OF  METALS. 


[Art.  97. 


Phcenix  Iron  in  Tension. 


POUNDS  STRESS    PER 
SQUARE  INCH. 

NUMBER 
OF    REPETITIONS. 

POUNDS  STRESS    PER 
SQUARE  INCH. 

NUMBER 
OF  REPETITIONS. 

From  o  to  52,800 
From  o  to  48,400 
From  o  to  44,000 
From  o  to  39,600 

800  rupture.  ' 
106,910  rupture. 
340,853  rupture. 
409,481  rupture. 

From          o  to  39,600 
From          o  to  35,200 
From  22,000  to  48,400 
From  26,400  to  48,400 

480,852  rupture. 
10,141,645  rupture. 
2»373i424  rupture. 
4,000,000  not  broken. 

Westphalia  Iron  in   Tension. 


From  o  to  52,800 

4,700  rupture. 

From  o  to  39,600 

180,800  rupture. 

From  o  to  48,400 

83,199  rupture. 

From  o  to  39,600 

596,089  rupture. 

From  o  to  48,400 

33,230  rupture. 

From  o  to  39,600 

433,572  rupture. 

From  o  to  44,000 

136,700  rupture. 

From  o  to  35,200 

280,121  rupture. 

From  o  to  44,000 

159,639  rupture. 

From  o  to  35,200 

566,344  rupture. 

Firth  &*  Sons'1  Steel  in   Tension. 


From  o  to  66,000 

83,319  rupture. 

From  o  to  55,000 

103,540  rupture. 

From  o  to  60,500 

168,396  rupture. 

From  o  to  53,900 

12,200,000  not  broken. 

From  o  to  55,000 

133,910  rupture. 

From  o  to  53,900 

229,230  rupture. 

From  o  to  55,000 

185,680  rupture. 

From  o  to  52,800 

692,543  rupture. 

From  o  to  55,000 

360,235  rupture. 

From  o  to  52,800 

12,200,000  not  broken. 

From  o  to  55,000 

186,005  rupture. 

From  o  to  50,600 

Krupp^s  Axle  Steel  in   Tension. 


From  o  to  88,000 
From  o  to  77,000 
From  o  to  66,000 
From  o  to  60,500 

18,741  rupture. 
46,286  rupture. 
170,000  rupture. 
123,770  rupture. 

From  o  to  55,000 
From  o  to  52,800 
From  o  to  50,600 

473,766  rupture. 
13,600,000  not  broken. 
12,200,000  not  broken. 

Art.  97.] 


EXPERIMENTAL  RESULTS. 


711 


Phosphor  Bronze  (unworkea)  in  Tension. 


POUNDS  STRESS    PER 
SQUARE   INCH. 

NUMBER 
OF   REPETITIONS. 

POUNDS   STRESS    PER 
SQUARE   INCH. 

NUMBER 
OF  REPETITIONS. 

From  o  to  27,500 
From  o  to  22,000 
From  o  to  16,500 

147,850  rupture. 
408,350  rupture. 
2,731,161  rupture. 

From  o  to  13,750 
From  o  to  13,750 

1,548,920  rupture. 
2,340,000  rupture. 

Phosphor  Bronze  (wrought)  in   Tension. 


From  o  to  22,000 
From  o  to  16,500 

53,900  rupture. 
2,600,000  not  broken. 

From  o  to  13,750 

1,621,300  rupture. 

Common  Bronze  in   Tension. 


From  o  to  22,000 
From  o  to  16,500 

4,200  rupture. 
6,300  rupture. 

From  o  to  u.ooo 

5,447,600  rupture. 

Phoenix  Iron  in  Flexure  (one  direction  only]. 


From  o  to  60,500 
From  o  to  55,000 
From  o  to  49,500 
From  o  to  44,000 

169,750  rupture. 
420,000  rupture. 
481,975  rupture. 
1,320,000  rupture. 

From  o  to  39,600 
From  o  to  35,200 
From  o  to  33,000 

4,035,400  ruptnre. 
3,420,000  rupture. 
4,820,000  not  broken. 

Westphalia  Iron  in  Flexure  (one  direction  only}. 


From  o  to  52,250 
From  o  to  49,500 
From  o  to  46,750 

612,065  rupture. 
457,229  rupture. 
799,543  rupture. 

From  o  to  44,000 
From  o  to  39,600 

1,493,511  rupture. 
31587.509  rupture. 

712 


FATIGUE  OF  METALS. 


[Art.  97. 


Homogeneous  Iron  in  Flexure  (one  direction  only}. 


POUNDS  STRESS    PER 
SQUARE  INCH. 

NUMBER 
OF  REPETITIONS. 

POUNDS  STRESS    PER 
SQUARE  INCH. 

NUMBER 
OF   REPETITIONS. 

From  o  to  60,500 
From  o  to  55,000 
From  o  to  49,500 
From  o  to  44,000 

169,750  rupture. 
420,000  rupture. 
481,975  rupture. 
1,320,000  rupture. 

From  o  to  39,600 
From  o  to  35,200 
From  o  to  33,000 

4,035,400  rupture. 
3,420,000  not  broken. 
48,200,000  not  broken. 

Firth  &*  Sons'  Steel  in  Flexure  (one  direction  only). 


From  o  to  63,250 
From  o  to  60,500 
From  o  to  55,000 

281,856  rupture. 
266,556  rupture. 
1,479,908  rupture. 

From  o  to  52,250 
From  o  to  49,500 
From  o  to  49,500 

578,323  rupture. 
5,640,596*  rupture. 
13,700,000  not  broken. 

*  Accidental. 

Krupp's  Axle  Steel  in  Flexure  (one  direction  only). 


From  o  to  77,000 
From  o  to  66,000 
From  o  to  60,500 

104,300  rupture. 
317,275  rupture. 
612,500  rupture. 

From  o  to  55,000 
From  o  to  55,000 
From  o  to  49,500 

729,400  rupture. 
1,499,600  rupture. 
43,000,000  not  broken. 

Krupp's  Spring  Steel  in  Flexure  (one  direction  only). 


From          o  to  110,000 

39,950  rupture. 

From  72,600  to  110,000 

19,673.300  not  broken. 

From         o  to    88,000 

117,000  rupture. 

From  66,000  to    99,000 

33,600,000  not  broken. 

From          o  to    66,000 

468,200  rupture. 

From  44,000  to    88,000 

35,800,000  not  broken. 

From         o  to    55,000 

40,600,000  not  broken. 

From  44,000  to    88,000 

38,000,000  not  broken. 

From         o  to    49.500 

32,942,000  not  broken. 

From  61,600  to    88,000 

36,000,000  not  broken. 

From  88,000  to  132,000 

35,600,000  not  broken. 

From  27,500  to    77,000 

36,600,000  not  broken. 

From  99,000  to  132,000 

33,478,700  not  broken. 

From  33,000  to    77,000 

31,152,000  not  broken. 

Art.  97.] 


EXPERIMENTAL  RESULTS. 


713 


Phosphor  Bronze  in  Flexure  (one  direction  only). 


POUNDS  STRESS   PER 
SQUARE   INCH. 

NUMBER 
OF  REPETITIONS. 

POUNDS  STRESS    PER 
SQUARE  INCH. 

NUMBER 
OF  REPETITIONS. 

From  o  to  22,000 
From  o  to  19,800 

862,980  rupture. 
8,151,811  rupture. 

From  o  to  16,500 
From  o  to  13,200 

5,075,169  rupture. 
10,000,000  not  broken. 

Common  Bronze  in  Flexure  (one  direction  only). 


From  o  to  22,000 
From  o  to  19,800 

102,659  rupture. 
151,310  rupture. 

From  o  to  16,500 
From  o  to  13,200 

837,760  rupture. 
10,400,000  not  broken. 

Pho3nix  Iron  in  Torsion  (both  directions). 


-  35,200  to    +  35)200 
-  33,000  to    +  33,000 
-  28,600  to    +  28,600 
-  26,400  to    +  26,400 

56,430  rupture. 
99,000  rupture. 
479,490  rupture. 
909,810  rupture. 

-  24,200  to    +  24,200 
—  22,000  to   +  22,000 
-  19,800  to    +  19,800 
-  17,600  to   +  17,600 

3,632,588  rupture. 
4,917,992  rupture. 
19,186,791  rupture. 
132,250,000  not  broken. 

English  Spindle  Iron  in  Torsion  (both  directions). 


-  37,400    tO     +  37,400 

204,400  rupture. 

-  30,800  to    +  30,800 

979,100  rupture. 

-  37,400  to    +  37,400 

147,800  rupture. 

-  28,600  to    +  28,600 

1,142,600  rupture. 

-  35,200  to    +  35,200 

911,100  rupture. 

-  28,600  to   +  28,600 

595,910  rupture. 

-  35,200  to    +  35,200 

402,900  rupture. 

-  26,400  to    +  26,400 

3,823,200  rupture. 

-  33,000  to    +  33,000 

1,064,700  rupture. 

-  26,400  to    +  26,400 

6,100,000  not  broken. 

-  33,000  to    +  33,000 

384,800  rupture. 

—  22,000  to    •+•  22,000 

8,800,000  not  broken. 

-  30,800  to    +  30,800 

Ii337'7°°  napture. 

-    22.000    tO      +    22,000 

4,000,000  not  broken. 

FATIGUE   OF  METALS. 


[Art.  97. 


Krupp's  Axle  Steel  in  Torsion  (both  directions'). 


POUNDS   STRESS    PER 

NUMBER. 

POUNDS   STRESS    PER 

NUMBER 

SQUARE  INCH. 

OF   REPETITIONS.   . 

SQUARE   INCH. 

OF   REPETITIONS. 

-  44,000   to    +  44,000 

367,400  rupture. 

-  46,200  to    -f  46,200 

55,100  rupture. 

-  39,600  to    +  39,600 

925,800  rupture. 

-  37,400  to    +  37,200 

797,525  rupture. 

-  37,400  to    +  37,40° 

4,900,000  not  broken. 

~  35i200  to    +  35j2oo 

1,665,580  rupture. 

-  35,200  to    +  35,200 

4,800,000  not  broken. 

-  33,000   to    +  33,000 

4^63,375  rupture. 

-  33,000   to    +  33,000 

5,000,000  not  broken. 

-  33,°°°    to    +  33,000 

45,050,640  rupture. 

In  Art.  33  will  be  found  some  experiments  by  the  late  Capt. 
Rodman,  U.  S.  A.,  on  the  fatigue  of  cast  iron,  but  they  are 
sufficient  in  number  and  character  to  show  the  general  effect 
only,  and  give  no  quantitative  results. 

The  specimens  used  in  all  the  preceding  experiments  were 
small. 

During  1860,  '61  and  '62,  Sir  Wm.  Fairbairn  constructed  a 
built  beam  of  plates  and  angles  with  a  depth  of  16  inches,  clear 
span  of  20  feet,  and  estimated  centre  breaking  load  of  26,880 
pounds. 

This  beam  was  subjected  to  the  action  of  a  centre  load  of 
6,643  pounds,  alternately  applied  and  relieved  eight  times  per 
minute  ;  596,790  continuous  applications  produced  no  visible 
alterations. 

The  load  was  then  increased  from  one-fourth  to  two- 
sevenths  the  breaking  weight,  and  403,210  more  applications 
were  made  without  apparent  injury. 

The  load  was  next  increased  to  two-fifths  the  breaking 
weight,  or  to  10,486  pounds;  5,175  changes  then  broke  the 
beam  in  the  tension  flange  near  the  centre. 

The  total  number  of  applications  was  thus  1,005,175. 

The  beam  was  then  repaired  and  loaded  with  10,500  pounds 
at  centre  158  times  ;  then  with  8,025  pounds  25,900  times,  and 


Art.  98.]  LAUNHARDT'S  FORMULA.  715 

finally  with  6,643  pounds  enough  times  to  make  a  total  of 
3,150,000. 

In  these  experiments  the  load  was  completely  removed 
each  time. 

It  is  thus  seen  that  vibrations  (without  shock)  with  one- 
fourth  the  calculated  breaking  centre  load  produced  no  appar- 
ent effect  on  the  resistance  of  the  beam,  but  that  two-fifths  of 
that  load  caused  failure  after  a  comparatively  small  number  of 
repetitions. 

It  is  probable  that  the  breaking  centre  load  was  calculated 
too  high,  in  which  case  the  ratios  \  and  f  should  be  somewhat 
increased^ 

Art.  98. — Formulae  of  Launhardt  and  Weyrauch. 

Let  R  represent  the  intensity  (stress  per  square  unit  of  sec- 
tion) of  ultimate  resistance  for  any  material  in  tension,  com- 
pression, shearing,  torsion  or  bending  ;  R  will  cause  rupture  at 
a  single,  gradual  application.  But  the  material  may  also  be 
ruptured  if  it  is  subjected  a  sufficient  number  of  times,  and 
alternately,  to  the  intensities  P  and  Q,  Q  being  less  than  P  and 
both  less  than  R,  while  all  are  of  the  same  kind.  When  Q  =  o 
let  P  —  W,  and  let  D  =  P  —  Q.  W  is  called  the  "  primitive 
safe  resistance,"  since  the  bar  returns  to  its  primitive  unstressed 
condition  at  each  application.  In  the  general  case  P  is  called 
the  "  working  ultimate  resistance." 

By  the  notation  adopted : 

P=  Q  +  D (l) 

But  by  Wohler's  law,  P  is  a  function  of  D ;  or, 

P  =  f(D) (2) 

A  sufficient  number  of  experiments  have  not  yet  been  made 


FATIGUE   OF  METALS.  [Art.  98. 

in  order  to  completely  determine  the  form  of  the  function 

f(D\ 

It  is  known,  however,  that  : 

For  0  =  o;     P  =  D  =  W\ 
and  for  "~7T^ 

D  =  o\    P=  Q  =  R. 

/ 

Provisionally,  Launhardt  satisfies  these  two  extreme  con- 
ditions by  taking  : 

R  -  W  r       R-  W  ,D 

p=D--(p-<Z    -    -    •    (3) 


Even  at  these  limits  this  is  not  thoroughly  satisfactory,  for 
n  D  =  o,  P  =  —  ( 
By  solving  Eq.  (3  : 


when  D  =  o,  P  =  —  (R  —  W),  or,  indeterminate. 


But  if  the  least  value  of  the  total  stress  to  which  any  mem- 
ber of  a  structure  is  subjected  is  represented  by  min  B,  and  its 

greatest  value  by  max  B,  there  will  result  mm    „  =  ^  . 

max  B       P 

Hence: 


P=  wi+-  (5) 

\  W       max  BJ 

which  is  Launhardt's  formula.  In  the  preceding  Article  some 
values  of  W  are  shown.  In  applying  Eq.  (5)  it  is  only  neces- 
sary to  take  the  primitive  safe  resistance,  W,  for  the  total 
number  of  times  which  the  structure  will  be  subjected  to  loads. 


Art.  98.]  WEYRAUCH'S  FORMULA. 

Since  bridges  are  expected  to  possess  an  indefinite  duration  of 
life,  in  such  structures  that  number  should  be  indefinitely 
large. 

Eq.  (5),  it  is  to  be  borne  in  mind,  is  to  be  applied  when  the 
piece  is  always  subjected  to  stress  of  one  kind,  or  in  one  direction 
only.  It  agrees  well  with  some  experiments  by  Wohler  on 
Krupp's  untempered  cast  spring  steel. 

If  the  stress  in  any  piece  varies  from  one  kind  to  another, 
as  from  tension  to  compression,  or  vice  versa  ;  or  from  one 
direction  to  another,  as  in  torsion  on  each  side  of  a  state  of  no 
stress,  Weyrauch  has  established  the  following  formula  by  a 
course  of  reasoning  similar  to  that  used  by  Launhardt. 

If  the  opposite  stresses,  which  will  cause  rupture  by  a  cer- 
tain number  of  applications,  are  equal  in  intensity,  and  if  that 
intensity  is  represented  by  5,  then  will  S  be  called  the  "  vibra- 
tion resistance  "  ;  this  was  established  by  Wohler  for  some 
cases,  and  some  of  its  values  are  given  in  the  preceding 
Article. 

Let  -j-  P  and  —  P'  represent  two  intensities  of  opposite 
kinds  or  in  opposite  directions,  of  which  P  is  numerically  the 
greater.  Then  if  D  =  P  +  P'  : 

P=  D  -  P'. 

The  two  following  limiting  conditions  will  hold  : 
For  P'  =  o  ;     P  =  D  =  W\ 
For/"  =  S      P  =  S  = 


But  by  Wohler's  law,  P  =  f(D),  and  the  two  limiting  con- 
ditions just  given  will  be  found  to  be  satisfied  by  the  pro- 
visional formula: 

D_      w-s    (p    p,        (6) 

^    ^      '          W 


zW-S-P 


71  8  FATIGUE  OF  METALS.  [Art.  98. 

By  the  solution  of  Eq.  (6)  : 


If,  without  regard  to  kind  or  direction,  max  B  is  numerically 
the  greatest  total  stress  which  the  piece  has  to  carry,  while 
max  B'  is  the  greatest  total  stress  of  the  other  kind  or  direc- 

tion, then  will  ^  =  ^?£j?  .     Hence,  there  will  result  the  fol- 
P        max  B 

lowing,  which  is  the  formula  of  Weyrauch  : 


P  =  W    ,  _  -  ....     (8) 

\  W      max  B  J 

Eqs.  (5)  and  (8)  give  values  of  the  intensity  P  which  are  to 
be  used  in  determining  the  cross  section  of  pieces  designed  to 
carry  given  amounts  of  stress.  If  n  is  the  safety  factor  and  F 
the  total  stress  to  be  carried,  the  area  of  section  desired  will  be  : 


-~P' 

~p  ^.  *1^f    J 

in  which  —  is  the  greatest  working  stress  permitted. 

If  for  wrought  iron  in  tension  W  =  30,000  and  R  =  50,000, 
Eq.  (5)  gives : 

„  /         2  min  B  \ 

P  =  30,000  [  i  H — 5 )  • 

\         3  max  B] 

Hence,  if  the  total  stress  due  to  fixed  and  moving  loads  in 
the  web  member  of  a  truss  is  -max  B  =  80,000  pounds,  while 
that  due  to  the  fixed  load  alone  is  min  B  =  40,000,  there  will 
result : 

/  9.         /in  nnn\ 

=   40,OOO. 


Art.  99.]  INFLUENCE   OF   TIME.  719 

In  such  a  case  the  greatest  permissible  working  stress  with 
a  safety  factor  of  3  would  be  about  13,300  pounds. 

For  steel  in  tension,  if  W  =  50,000  and  R  =  75,000: 


P  =  50,000  (i  +  I  ?™ 
\         2  ma 


max  B 

For  wrought  iron  in  torsion,  if  S  =  18,000  and  W  =  24,000, 
Eq.  (8)  will  give  : 

/          i  max 
P  =  24,000  (i 


\         4  max  B 

Other  methods  based  on  Wohler's  experiments  have  been, 
deduced  by  Muller,  Gerber  and  Schaffer,  of  which  synopses 
may  be  found  in  Du  Bois'  translation  of  Weyrauch's  "  Struct- 
ures of  Iron  and  Steel." 


Art.  99. — Influence  of  Time  on  Strains. 

In  the  section  "  elevation  of  ultimate  resistance  and  elastic 
limit"  in  Art.  32,  the  effect  of  prolonged  tensile  stress  and 
subsequent  rest  between  the  elastic  limit  and  ultimate  resist- 
ance, was  shown  to  be  the  elevation  of  both  those  quantities. 
It  is  a  matter  of  common  observation,  however,  that  if  a  piece 
of  wrought  iron  be  subjected  to  a  tensile  stress  nearly  equal  to 
its  ultimate  resistance,  and  held  in  that  condition,  that  the 
stretch  will  increase  as  the  time  elapses. 

Experiments  are  still  lacking  which  may  show  that  a  piece 
of  metal  can  be  ruptured  by  a  tensile  stress  much  below  its 
ultimate  resistance.  It  may  be  indirectly  inferred,  however, 
from  experiments  on  flexure,  that  such  failure  may  be  pro- 
duced, as  the  following  by  Prof.  Thurston  will  show. 

A  bar  10  parts  tin  and  90  parts  copper,  I  X  I  x  22  inches 
and  supported  at  each  end,  sustained  about  65  per  cent,  of  its 


720  FATIGUE  OF  METALS.  [Art.  99. 

breaking  load  at  the  centre  for  five  minutes.  During  that  time 
its  deflection  increased  0.021  inch.  The  same  bar  sustained 
1,485  pounds  at  centre  for  13  minutes  and  then  failed. 

A  second  bar  of  the  same  size,  but  90  parts  tin  and  10  parts 
copper,  was  loaded  at  the  centre  with  160  pounds,  causing  a 
deflection  of  1.294  inches.  After  10  minutes  the  deflection 
had  increased  0.025  inch;  after  one  day,  i.ooinch;  after  two 
days,  2.00  inches  ;  and  after  three  days,  3.00  inches,  when  the 
bar  failed  under  the  load  of  160  pounds. 

Another  bar  of  the  same  size  showed  remarkable  results ;  it 
was  composed  of  90  parts  zinc  and  10  parts  copper.  It  gave 
the  same  general  increase  of  deflection  with  time,  but  eventually 
JDroke  under  a  centre  load  which  ran  down  from  1,233  to  911 
pounds,  after  holding  the  latter  about  three  minutes. 

A  bar  of  the  same  size  and  96  parts  copper  with  4  parts  tin, 
after  it  had  carried  700  pounds  at  centre  for  sixty  minutes  was 
loaded  with  1,000  pounds,  with  the  following  results  : 

AFTER  DEFLECTION. 

o  minute 3.118  inches. 

5  minutes 3-54O  " 

15  minutes 3.660  " 

45  minutes 4. 102  " 

75  minutes . ....  7.634  " 

Broke  under  1,000  pounds. 

A  wrought-iron  bar  of-  the  same  size  gave,  under  a  centre 
load  of  1,600  pounds  : 

AFTER  DEFLECTION. 

O  minute 0.489  inch. 

3  minutes 0.632     " 

6  minutes 0.650     " 

16  minutes 0.660     " 

344  minutes 0.660     " 

It  subsequently  carried  2,589  pounds  with  a  deflection  of 
4.67  inches. 


Art.  99.]  INFLUENCE   OF   TIME.  721 

During  1875  and  1876  Prof.  Thurston  made  a  number  of 
other  similar  experiments  with  the  same  general  results. 

Metals  like  tin  and  many  of  its  alloys  showed  an  increasing 
rate  of  deflection  and  final  failure,  far  below  the  so-called 
"  ultimate  resistance."  The  wrought-iron  bars,  however,  showed 
a  decreasing  increment  of  deflection,  which  finally  became  zero, 
leaving  the  deflection  constant. 

Whether  there  may  be  a  point  for  every  metal,  beyond 
which,  with  a  given  load,  the  increment  of  deflection  may 
retain  its  value  or  go  on  increasing  until  failure,  and  below 
which  this  increment  decreases  as  the  time  elapses,  and  finally 
becomes  zero,  is  yet  undetermined,  but  seems  probable. 

It  does  not  follow,  therefore,  that  the  principle  enunciated 
in  the  section  named  at  the  beginning  of  this  Article,  is  to  be 
taken  without  qualification.  If  "  rest  "  under  stress,  too  near 
the  ultimate  resistance,  be  sufficiently  prolonged,  it  has  been 
seen  that  it  is  possible  that  failure  may  take  place. 

In  verifying  some  experimental  results  by  Herman  Haupt, 
determined  over  forty  years  ago,  Prof.  Thurston  tested  three 
seasoned  pine  beams  about  i^  inches  square  and  40  inches 
length  of  span,  and  found  that  60  per  cent,  of  the  ordinary 
"breaking  load"  caused  failure  at  the  end  of  8,  12  and  15 
months.  In  these  cases  the  deflection  slowly  and  steadily  in- 
creased during  the  periods  named. 

Two  other  sets  of  three  pine  beams  each,  broke  under  80 
and  95  per  cent,  of  the  usual  "  breaking  load,"  after  much 
shorter  intervals  of  time. 

In  all  these  instances  it  is  evident  that  the  molecules  under 
the  greatest  stress  "  flow  "  over  each  other  to  a  greater  or  less 
extent.  In  the  cases  of  decreasing  increments  of  strain,  the 
new  positions  afford  capacity  of  increased  resistance  ;  in  the 
others,  those  movements  are  so  great  that  the  distances  be- 
tween some  of  the  molecules  exceed  the  reach  of  molecular 
action,  and  failure  follows. 

In  many  cases  strained  portions  of  material  recover  partially 
46 


722  FATIGUE  OF  METALS.  [Art.  99. 

or  wholly  from  permanent  set.  In  such  cases  a  portion  of  the 
material  has  been  subjected  to  intensities  of  stress  high  enough 
to  produce  true  "  flow  "  of  the  -molecules,  while  the  remaining 
portion  has  not.  The  internal  elastic  stresses  in  the  latter  por- 
tion, after  the  removal  of  the  external  forces,  produce  in  time 
a  reverse  flow  in  consequence  of  the  elastic  endeavor  to  resume 
the  original  shape. 

It  is  altogether  probable  that  the  phenomena  of  fatigue  and 
flow  of  metals  are  very  intimately  associated.  Some  of  the 
prominent  characteristics  of  the  latter  will  be  given  in  the  next 
chapter.- 


CHAPTER    XIV. 

THE  FLOW  OF  SOLIDS. 

Art.   100. — General   Statements. 

ALTHOUGH  there  is  no  reason  to  suppose  that  true  solids 
may  not  retain  a  definite  shape  for  an  indefinite  length  of  time 
if  subjected  to  no  external  force  other  than  gravity,*  many 
phenomena  resulting  both  from  direct  experiment  for  the  pur- 
pose, and  incidentally  from  other  experiments  involving  the 
application  of  external  stress  of  considerable  intensity,  show 
that  a  proper  intensity  of  internal  stress  (in  many  cases  com- 
paratively low)  will  cause  the  molecules  of  a  solid  to  flow,  at 
ordinary  temperatures,  like  those  of  a  liquid.  And  this  flow, 
moreover,  is  entirely  different  from,  and  independent  of,  the 
elastic  properties  of  the  material  ;  for  it  arises  from  a  perma- 
nent and  considerable  relative  displacement  of  the  molecules. 
Nor  is  it  to  be  confounded  with  that  internal  "  friction  "  which, 
if  an  elastic  body  is  subjected  to  oscillations,  causes  the  ampli- 
tudes to  gradually  decrease  and  finally  disappear,  even  in 
vacuo.  This  latter  motion  is  typically  elastic  and  the  retarding 
cause  may  be  considered  a  kind  of  elastic  friction. 

It  is  evident  that  if  a  mass  of  material  be  enclosed  on  all  its 
faces,  or  outer  surfaces,  but  one  or  a  portion  of  one,  and  if 
external  pressure  be  brought  to  bear  on  those  faces,  the  mate- 
rial will  be  forced  to  move  to  and  through  the  free  surface  ;  in 

*  This,  perhaps,  may  be  considered  a  definition  of  a  true  solid. 


724 


FLOW  OF  METALS. 


[Art.  100. 


other  words,  the  flow   of  the  material  will  take  place  in   the 

direction  of  least  resistance. 

The  theory  of  the  flow  of  solids 
to  be  given  is  that  developed  by 
Mons.  H.  Tresca  in  his  "  Memoire 
sur  1'Ecoulement  des  Corps  So- 
lides,"  1865.  He  made  a  large 
number  of  experiments  on  hard 
and  soft  metals,  ceramic  pastes, 
sand  and  shot. 

These  different  materials  all 
manifested  the  same  characteristics 
of  flow,  which  are  well  shown  in 
Fig.  2.  ABCD,  Fig.  I,  is  supposed 
to  be  a  cylindrical  mass  of  Mead 
with  circular  horizontal  section,  con- 
fined in  a  circular  cylinder,  MN, 
closed  at  one  end  with  the  excep- 
tion of  the  orifice  O. 

This  cylinder  is  supported  on 
the  base  7W,  while  the  face  AB  of 
the  lead  receives  external  pressure 
from  a  close-fitting  piston.  When 
the  pressure  is  sufficiently  increased, 
the  face  AB  in  Fig.  I  sinks  to  AB 
in  Fig.  2,  while  the  column  hkHK, 
in  the  latter  figure,  is  forced  to  flow 
through  the  orifice  O. 

In  Tresca's  experiments  with 
lead,  the  diameter  AB  was  about 
3.9  inches  ;  the  diameter  HK  of  the 


Fig.2 


H 


M— 


Fig.3 


orifice,  from  0.75  in.  to  1.5  ins.,  while  the  length  of  the  column 
or  jet  hK  varied  from  0.4  in.  to  about  24  ins.  The  total  press- 
ure on  the  face  AB  varied  from  119,000  to  198,000  pounds. 
The  initial  thickness  AD  varied  from  0.24  inch  to  2.4  inches. 


Art.  1 01.]  HYPOTHESES  OF   TRESCA.  ?2$. 

Some  experiments  exhibiting  in  a  remarkably  clear  manner 
the  flow  of  metals  in  cold  punching  were  made  by  David 
Townsend  in  1878,  and  the  results  were  given  by  him  in  the 
"  Journal  of  the  Franklin  Institute  "  for  March  of  that  year. 
If  the  dotted  rectangle  ABFG,  Fig.  3,  shows  the  original  out- 
line of  the  middle  section  of  a  nut  before  punching,  he  found 
that  the  final  outline  of  the  same  section  would  be  represented 
by  the  full  lines.  The  top  and  bottom  faces  were  depressed 
by  the  punching,  as  shown  ;  the  upper  width  AB  remained 
about  the  same,  but  the  lower,  GFy  was  increased  to  CD.  Al- 
though the  depth  of  the  nut,  AC,  was  1.75  inches,  the  length 
of  the  core  punched  out  was  only  1.063  inches.  The  density  of 
this  core  was  then  examined  and  found  to  be  the  same  as  that 
of  the  original  nut.  Hence  a  portion  of  the  core  equal  in 
length  to  1.75  —  1.063  =  0.687  inch  was  forced,  or  flowed,  back 
into  the  body  of  the  nut.  Subsequent  experiments  showed 
that  this  flow  did  not  take  place  at  the  immediate  upper  sur- 
face AB,  nor  very  much  in  the  lower  half  of  the  nut,  but  that 
it  was  chiefly  confined  to  a  zone  equal  in  depth  to  about  half 
that  of  the  nut,  the  upper  surface  of  which  lies  a  very  short 
distance  below  the  upper  face  of  the  nut.  The  location  of  this 
zone  is  shown  by  the  lines  HK  and  MN  in  Fig.  3. 

Tresca's  experiments  on  punching  showed  essentially  the 
same  result. 


Art.  101. — Tresca's  Hypotheses. 

The  central  cylinder  FGKH,  Fig.  I  of  Art.  TOO  was  called 
by  Tresca  the  "  primitive  central  cylinder."  As  the  metal 
flows,  this  cylinder  will  be  drawn  out  into  the  volume  of  revo- 
lution, whose  axis  is  that  of  the  orifice  and  whose  meridian 
section  is  FGkKHk,  Fig.  2,  the  diameter  FG  being  gradually 
decreased. 

It  was  found  by  experiment  that   if  the  original  mass  AC, 


726  FLOW  OF  SOLIDS.  [Art.  IOI. 

Fig.  i,  was  composed  of  horizontal  layers  of  uniform  thickness, 
the  reduced  mass  in  Fig.  2  was  also  composed  of  the  same 
number  of  layers  of  uniform  thickness,  except  in  the  immediate 
vicinity  of  the  central  cylinder. 

Tresca  then  assumed  these  three  hypotheses  : 
i°. —  The  density  of  the  material  remains  the  same  whether  in 
the  cylinder  or  in  the  jet ;  in  other  words,  the  volume   of  the 
material  in  the  jet  and  in  the  cylinder  remains  constant. 

Let  R  =  radius  of  the  cylinder. 

Let  Rt  =  radius  of  the  orifice. 

Let  y  =  variable  length  of  the  jet  (i.  e.,  hH}. 

Let  D  =  original  depth  of  material  (BC  =  AD,  Fig.  i) 

in  the  cylinder. 
Let  d  —  variable  depth  of  material  (BC  —AD,  Fig.  2) 

in  the  cylinder. 

Then  by  the  hypothesis  just  stated  : 

R*d  -  R*D  -  R?y     .      .....     (i) 

2°.  The  rate  of  compression  along  any  and  all  lines  parallel 
to  the  axis  of  the  primitive  central  cylinder,  and  taken  outside  of 
that  limit,  is  constant. 

If,  then,  the  material  lying  outside  of  the  central  cylinder 
be  divided  into  horizontal  layers  of  equal  thickness,  a  very 
small  decrease  in  the  variable  depth  equal  to  d(a)  will  cause 
the  same  amount  of  material  to  move  or  flow  from  each  of 
these  layers  into  the  space  originally  occupied  by  the  central 
cylinder,  thus  causing  a  portion  of  the  material  previously 
resting  over  the  orifice  to  flow  through  the  latter.  If  d(d)  is 
the  indefinitely  small  change  of  depth,  and  dR^  the  indefinitely 
small  change  in  the  radius  of  the  cylindrical  portion  resting 
over  the  orifice,  then  the  equality  of  volumes  expressing  this 
hypothesis  is  the  following  : 


Art.   102.]  MERIDIAN  SECTION.  727 


n(E?  -  R,2}   .   d(d)  = 
or  : 

d(d)  =     2R,  dR, 
d    ~  R3  -  R- 


(2) 


3°.  —  The  rate  of  decrease  of  the  radius  of  the  primitive  cen- 
tral cylinder  is  constant  throughout  its  length  at  any  given  instant 
during  flow. 

Let  r  be  any  radius  less  than  R19  then  if  the  latter  is  de- 
creased by  the  very  small  amount  dR»  the  former  will  be 
shortened  by  the  amount  dr  ;  and  by  the  last  hypothesis  there 
must  result  : 


,.N 


This  is  a  perfectly  general  equation,  in  which  r  may  or  may 
not  be  the  variable  value  of  the  radius  of  that  portion  of  the 
primitive  central  cylinder  remaining  above  the  orifice  at  any 
instant  during  flow. 

These  are  the  three  hypotheses  on  which  Tresca  based  his 
theory  of  the  flow  of  solids.  It  is  thus  seen  to  be  put  upon  a 
purely  geometrical  basis,  entirely  independent  of  the  elastic  or 
other  properties  of  the  material. 


Art.  102.— The  Variable  Meridian  Section  of  the  Primitive  Central 

Cylinder. 

The  meridian  curve  haH,  or  kbK,  Fig.  2  of  Art.  100,  may 
now  easily  be  determined. 

Eq.  (i)  of  Art.  101  may  take  the  first  of  the  following  forms, 
while  its  differential,  considering  d  and  y  variable,  may  take 
the  second  : 


728  FLOW  OF  SOLIDS.  [Art.  IO2. 


Dividing  the  second  by  the  first  : 


d(d)  _  dy  2R,  dR, 

d  &    _  ""  R*  -  R*  ' 


The  last  member  of  this  equation  is  simply  Eq.  (2)  of  Art. 
101  ;  and  if  the  value  of  dR»  in  Eq.  (3)  of  the  same  Article, 
be  inserted  in  the  third  member  of  this  equation,  there  will 
result  : 

2R*          dr  dy 


•R 


Integrating  between  the  limits  of  r  and  Rv  and  remember- 
ing that  r  will  be  restricted  to  the  representation  of  the  radius 
of  that  portion  of  the  primitive  central  cylinder  which  remains, 
at  any  instant,  over  the  orifice,  by  taking  y  —  o  for  r  —  R^  : 


2R* 


"log"  indicates  a  Napierian  logarithm. 

Passing  from  logarithms  to  the  quantities  themselves,  and 

rt*r\ iir^inrr   • 


reducing  : 


Art.  103.]  HORIZONTAL   SECTIONS.  729 


This  is  the  desired  equation  of  the  line,  in  which  r  is  meas- 
ured normal  to  the  axis  of  the  cylinder  or  jet,  while  y  is  meas- 
ured along  that  axis  from  the  extremity  of  the  jet.  When  the 
material  is  wholly  expelled  : 

y  —  -FTI  D,    and     r  —  o. 

Eq.  (2)  is  applicable  to  the  jet  only.  For  the  line  hF  or  Gk, 
resort  will  had  to  the  equation  : 

d(d)  _        2R?      dr 
d      "  R2  —  R?    r  ' 

Again  integrating  between  the  limits  d  and  Z>,  or  r  and  R» 
and  reducing  : 

r~R,(£f*P~. (2) 

This  value  of  r  is  the  radius  of  that  portion  of  the  primitive 
central  cylinder  which  remains  over  the  orifice  when  D  is  re- 
duced to  d. 


Art.  103.— Positions  in  the  Jet  of  Horizontal  Sections  of  the  Primitive 

Central    Cylinder. 

That  portion  of  the  primitive  central  cylinder  below  ab  in 
Fig.  i  of  Art.  100,  will  be  changed  to  abKH  in  Fig.  2  of  the 
same  Article. 

If,   in  the  latter  Fig.,  y  is  the  distance  from  HK  to  ab, 


730 


FLOW  OF  SOLIDS.  [Art,  103. 


measured  along  the  axis,  then  the  volume  of  HKab  will  have 
the  value 

I    nr*  dy. 

Jo 

If  d'  is  the  distance  aF—bG,  in  Fig.  i,  the  equality  of 
volumes  will  give  : 

[y'r*  dy  =  R*(D  -  d'). 

Jo 

Eq.  (i)  of  Art.  102  gives  : 


R*- 


If  TV  is  the  number  of  horizontal  layers  required  to  compose 
the  total  thickness  D,  and  n  the  number  in  the  depth  d'  : 


D       N 
Hence  : 


Art.  105.]  PATH  OF  MOLECULE.  731 

Tresca  computed  values  of  y  for  some  of  his  experiments, 
and  compared  the  results  with  actual  measurements.  The 
agreement,  though  not  exact,  was  very  satisfactory.  Within 
limits  not  extreme,  the  longer  the  jet  the  more  satisfactory  was 
the  agreement. 


Art.  104.— Final  Radius  of  a  Horizontal  Section  of  the  Primitive  Central 

Cylinder. 

Let  it  be  required  to  determine  what  radius  the  section 
situated  at  the  distance  d'  from  the  upper  surface  of  the  primi- 
tive central  cylinder  will  possess  in  the  jet. 

It  will  only  be  necessary  to  put  for  y  in  Eq.  (i)  of  Art.  102, 
the  value  of/  taken  from  Eq.  (i)  of  Art.  103.  This  operation 
gives  : 


Hence  : 


d'\     *& 

r  =RI\D)         .......  (0 

If  RI  is  small,  as  compared  with  R,  there  will  result  ap- 
proximately : 


w 


Art.  105.—  Path  of  any  Molecule. 


The  hypotheses  on  which  the  theory  of  flow  is  based  enable 

the  hypothetical  path  of  any  molecule  to  be  easily  established. 

In  consequence  of  the  nature  of  the  motion  there  will  be 


732  FLOW  OF  SOLIDS.  [Art.   105. 

three  portions  of  the  path,  each  of  which  will  be  represented 
by  its  characteristic  equation,  as  follows  : 

First  :  let  the  molecule  lie  outside  of  the  primitive  central 
cylinder. 

Let  R'  and  H  be  the  original  co-ordinates  of  the  molecule 
considered,  measured  normal  to  and  along  the  axis  of  the 
cylinder,  respectively,  from  the  centre  of  the  orifice  HK(¥'\g.  I 
Art.  100)  as  an  origin,  while  r  and  h  are  the  variable  co- 
ordinates. 

The  first  hypothesis,  by  which  the  density  remains  con- 
stant, then  gives  the  following  equation  : 


-  R2)H  =  n(R2  - 
or  : 

hR2  -  hr2  =  (R2  -  R2)H (i) 

This  is  the  equation  to  the  path  of  the  molecule,  in  which 
r  must  always  exceed  Rv 

As  this  equation  is  of  the  third  degree,  the  curve  cannot  be 
one  of  the  conic  sections. 

Second  :  let  the  molecule  move  in  the  space  originally  occupied 
by  the  central  cylinder. 

While  h  and  r  now  vary,  the  volume  nr2(D  —  h]  must  re- 
main constant.  When  r  =  Rt  let  h  =  h^.  Hence  : 


h)  =  R2(D-k^      .....     (2) 
But  if  h  =  7/x  and  r  =  R^m  Eq.  (i)  : 


Placing  this  value  in  Eq.  (2)  : 

-  k)  =  R?D  -H  ?L  .     ...    (3) 


Art.   IO5.]  PATH  OF  MOLECULE.  733 

Third  :  let  the  molecule  move  in  the  jet.  % 

After  the  molecule  passes  the  orifice,  its  path  will  evidently 
be  a  straight  line  parallel  to  the  axis  of  the  jet.  Its  distance 
rx  from  that  axis  will  be  found  by  putting  h  =  o  in  Eq.  (3). 
Hence  : 

H  R2  -  R 


ADDENDA. 


Addendum  to  Art.  20. 

SOME  problems  similar  to  that  treated  on  pages  134  and 
135,  but  of  a  less  simple  character,  arise  in  connection  with  the 
design  of  railway  track  stringers.  The  general  method  of  solu- 
tion of  such  problems  has  already  been  indicated  in  Art.  20, 
but  will  be  here  applied  to  four  equal  weights,  each  being 
represented  by  W. 

Case  I. 

The  relative  position  of  these  four  weights  is  shown  in  Fig. 
I,  in  which  /  is  the  span  and  a,  c  and  b  the  distances  separating 
the  adjacent  pairs  of  weights.  The  latter  distances  are  fixed 
or  constant. 


Fig.l 


Since  c  and  b  are  each  greater  than  a,  the  shear  will  be 
zero  under  the  weight  A  when  all  the  weights  are  so  placed  as 


Art.  20.]  ADDENDA,  735 

to  give  the  beam  its  greatest  possible  bending  moment.  It 
will  only  be  necessary,  then,  to  find  such  a  position  of  the 
loading  as  will  make  the  bending  moment  under  A  the  greatest 
possible. 

Let  x  be  measured  to  the  left  from  the  right  abutment,  as 
shown ;  then  the  left  reaction  at  R  will  be  : 


=  W  (- 


Hence  the  bending  moment  under  A  will  be  : 


Taking  the  first  derivative  in  reference  to  x 


Inserting  this  value  of  x  in  Eq.  (2)  and  indicating  the  re- 
sulting maximum  moment  by  AfI  : 


MT  is  the  greatest  bending  moment  to  which  the  beam  or 
stringer  can  ever  be  subjected,  and  it  will  be  found  at  the  dis- 
tance (l-x-a-c)  =  (#l-}6a-%c+  Yzti)  from  either 
abutment. 


ADDENDA.  [Art.  2O. 


But  in  Fig.  I,  if  x^  is  the  distance  from  the  right  abutment 
to  the  centre  of  gravity  of  the  entire  load  : 


(5) 


Hence,  the  centre  of  span  is  midway  between  the  centre  of 
gravity  of  the  load  and  wheel  A  or  point  of  greatest  bending. 


Case  2. 

In  this  case  let  a  =  c.     Making  this  change  in  Eq.  (4),  the 
value  of  the  greatest  moment  becomes  : 


w 


The  distance  of  greatest   bending   from    either  abutment 
takes  the  value  : 


(7) 


Case  3. 

Let  a  =  c  =  b.     Ml  and  ^  then  take  the  values  : 


--  Wa 


(9) 


In  the  case  of  an  actual  stringer,  the  equal  weights  Ware 
the  weights  on  the  driving  wheels  of  a  locomotive. 


Art.  75.]  ADDENDA.  737 


Addendum  to  Art.  73. 
• 

A  butt  joint  with  a  set  of  single  or  double  cover  plates  or 
butt  straps  may  be  formed  in  such  a  manner  that  the  rivets 
and  cover  plates  will  take  very  nearly  or  exactly  their  proper 
proportional  loads.  Each  set  of  cover  plates  is  composed  of 
a  series  uniformly  decreasing  in  length,  the  longest  of  the 
series  lying  adjacent  to  the  main  plates  or  members  joined. 
One  row  of  rivets  parallel  to  the  joint  is  then  put  through  each 
end  of  each  cover  plate,  and,  of  course,  also  through  those 
lying  underneath.  In  this  manner  the  number  of  rivets  from 
the  end  of  the  longest  or  lowest  cover  plate  to  any  section 
parallel  to  the  joint  is  proportional  to  the  sectional  area  of  the 
covers  against  which  they  pull ;  the  joint  is  consequently  of 
nearly  uniform  resistance. 

The  number  of  butt  straps  or  cover  plates  in  a  set  depends 
upon  the  size  of  the  members  joined. 

In  most  cases  the  rivets  cannot  take  exactly  their  propor- 
tional loads,  for  the  reason  that  those  portions  of  the  members 
joined  which  lie  within  the  limits  of  the  joint  are  not  of  uni- 
form resistance,  as  the  system  of  covers  is. 


Addendum  to  Art.  75. 

Some  recent  experiments  (March,  1883),  on  steel  eye-bars 
manufactured  by  the  Edge  Moor  Iron  Co.  under  their  patents, 
and  tested  to  failure,  show  such  interesting  and  successful  re- 
sults that  they  should  be  noticed  in  a  work  of  this  character. 
The  heads  were  of  the  general  form  shown  on  page  647,  the 
front  portion  ABD  having  been  described  with  the  radius  r  = 

they  were  not  thickened. 

The  following  are  the  data  and  results  of  tests : 
47 


733 


ADDENDA. 


[Art.  75. 


/      —  length  between  centres  of  pin  holes  ; 

d      =  diameter  of  pin  ; 

D     =  A  D,  Fig.  3,  of  page  647  ;  or  twice  CB  ; 

n      —  ratio  of  pin  diameter  over  width  of  bar  ; 

e      —  percentage    of  excess  of  eye  section  AD  over 

area  of  bar  section  ; 

E.  L.  =  elastic  limit  in  pounds  per  sq.  in.  ; 
Ult.     —  ultimate  resistance  in  pounds  per  sq.  in. ; 

q      =  percentage  of  reduction  of  fracture  section  ; 

/      =  percentage  of  stretch  of  original  (/  —  d\ 

All  bars  were  3  X  {£  =  2.4375  scl-   ms-   'm  sectional  area, 
and  were  of  mild  steel. 


NO. 

/. 

d. 

D. 

«. 

e. 

E.  L. 

Ult. 

9" 

P- 

Ft.    Ins. 

Ins. 

Ins. 

I 

5        o 

3tt 

81 

1-31 

57-2 

48,000 

71,400 

48.0 

13-8 

2 

5        3 

4f«r 

81 

1.50 

4i.5 

49,000 

69,600 

48.6 

13.7 

3 

5        f 

4$ 

81 

1.50 

40.0 

45,400 

68,800 

43-7 

14.8 

4 

5        f 

3,3. 

7i 

i.  06 

51-6 

44,100 

64,  700 

44-0 

17.0 

5 

5      A' 

3:4 

7l 

1.14 

38.0 

44,100 

70,  900 

54-o 

8.0 

6 

5      1% 

sH 

7J 

1.23 

33-2 

44,100 

73,000 

50.6 

7-5 

7 

4     i|& 

4ti 

9i 

1.64 

55-5 

49?4oo 

70,100 

48.0 

15.8 

8 

4     itt 

5& 

9^ 

1.72 

46.4 

44,100 

64,  700 

40.8 

16.5 

9 

4     i|3 

5tl 

9l 

1.89 

29.2 

42,000 

64,400 

42.0 

17-5 

10 

4     ill 

3H   5fr 

71    9l 

1.2     1.8 

33.0  40.0 

44,  100 

64,400 

48.0 

16.3 

All  values  in  column  e,  except  those  for  No.  10,  aVe  means 
of  two  (one  for  each  end  of  the  bar),  but  in  no  case  did  either 
of  the  latter  vary  more  than  one  and  one  half  per  cent,  from 
the  mean. 

The  pin  holes  were  elongated  from  one  quarter  to  one  inch. 

All  bars  broke  in  the  body  of  the  bar,  and  none  nearer  the 
centre  of  the  eye  than  about  ten  inches.  Half  of  them  broke 
in  the  vicinity  of  the  centre  of  the  bar. 

With  the  exception  of  No.  4  it  will  be  observed  that  e  is 


Art.  75.]  ADDENDA.  739 

much  less  than  required  by  ordinary  practice  with  iron  bars. 
But  it  will  also  be  observed  that  n  is  much  larger  than  is  usually 
found  in  bridge  practice.  With  smaller  values  of  n,  larger  ex- 
cesses (e)  might  be  found  necessary,  as  No.  4  would  seem  to 
indicate.  In  any  event,  however,  the  tests  show  that  the 
homogeneous  character  of  steel  insures  it  against  much  of  the 
injury  which  iron  suffers  in  the  upsetting  orocess  preceding 
the  formation  of  the  head. 


INDEX. 


A. 

Actual  energy  of  elasticity 95,  96 

Adhesion  between  bricks  and  mortar 364,  365 

Alloys  of  copper,  tin  and  zinc  in  compression 387-389 

Alloys  of  copper,  tin  and  zinc  in  tension 336,  339-341,  343,  346,  347 

Aluminium  bronze  in  tension 343 

American  Bridge  Co.  column 438,  439,  442,  448 

Angle  iron  column 445 

Angle  irons  as  columns 475~477 

Annealing,  effect  of,  on  wrought  iron 245 

Annealing,  effect  of,  on  steel 298,  313,  319,  331 

Artificial  stones  in  compression 393,  395,  396 

Artificial  stones  in  tension 363,  364 


B. 

Bauschinger's  experiments  on  steel 333 

Bauschinger's  experiments  on  wrought  iron 262-268 

Beam,  continuous 177 

Beam,  one  end  fixed  and  other  simply  supported 182-184 

Beam,  fixed  at  both  ends 188-190 

Beam,  non-continuous  with  uniform  load 139,  174 

Beam,  non-continuous  with  single  weight 138,  174 

Beams,  solid 514,  515 

Beams,  solid,  cast  iron 518-520 

Beams,  solid,  cement,  mortar  and  concrete 537-542 

Beams,  solid,  combined  iron  and  steel 523,  524 

Beams,  solid,  copper,  tin,  zinc  and  alloys ,  524-526 

Beams,  solid,  practical  formulae  for 543,  544 

Beams,  solid,  steel 520-523 


742  INDEX. 


PAGE 

Beams,  solid,  stone 543 

Beams,  solid,  timber 526-536 

Beams,  solid,  wrought  iron 515-518 

Bending  by  continuous  normal  load 679-680 

Bending  by  oblique  forces 674-679 

Bending  in  riveted  joints 609-612,  614,  615 

Bending  moments 129,  132,  133 

Bending  moments,  greatest 137 

Bending  moments,  greatest  or  least 134 

Bending  moments,  greatest,  with  four  weights 735,  736 

Bessemer  steel,  coefficient  of  elasticity 288,  290,  291 

Bessemer  steel,  ultimate  resistance 301-303,  321 

Box  beams 595,  596 

Bouscaren's  experiments  (columns) 446,  447,  453 

Brass,  fine  yellow,  in  compression 389 

Brass,  in  tension 336,  340-344 

Brass,  red 350 

Brick  in  compression 397 

Brick  in  tension ., 364 

Buckling  of  latticed  columns 458,  459 

Building  stones,  natural,  in  compression 398-402 

Built  beams,  steel 600,  601 

Built  beams,  wrought  iron 578-601 

Bulging  of  plates 665-674 

Butt  joints,  double  covers 606,  633-639 

Butt  joints  of  uniform  resistance 737 

Butt  joints,  single  butt  strap.     See  "  Lap  Joint." 


C.  i 

Cables  or  ropes 648-654 

Cadmium 351,  352 

Cantilever 132 

Cantilever  with  single  load 1 72 

Cantilever  with  uniform  load 142,  1 72 

Cast-iron  columns,  hollow  round 470,  472 

Cast-iron  columns,  solid  round 447,  470,  472 

Cast-iron  flanged  beams 554-559 

Cast  iron  in  compression 376 

Coefficients  of  elasticity 377 

Ultimate  resistance 378,  379 

Effect  of  remelting 379 


INDEX.  743 


PAGE 

Cast  iron  in  shearing 487-489,  493 

Cast  iron  in  tension 276 

Coefficient  of  elasticity  and  elastic  limit 276 

Ultimate  resistance 279 

Effect  of  remelting 282 

Effect  of  continued  fusion 284 

Repetition  of  stress 284 

Effect  of  high  temperatures 286,  347 

Cast  iron  in  torsion , 487,  500-503 

Cast  steel,  ultimate  tensile  resistance 301,  302 

Cement  in  compression 39°~392 

Cement  mortar  in  compression 354,  39°-392 

Cement  mortar  in  tension 354,  359,  360 

Portland 360 

Various  brands 354 

Cement,  pure,  in  tension 353 

Variation  of  strength  with  age 355,  356,  360,  361 

Maclay's  experiments 357*  359 

Grant's  experiments 361 

Keene's  cement 361 

Parian  cement 361 

Portland  cement 357>  359>  3«o 

Chain  cables 652-654 

Channels  as  columns 455>  46l,  462 

Chemical  constitution  of  wrought  iron 270 

Chrome  steel,  coefficient  of  elasticity 286 

Circular  cylinder,  torsion  of 75.  76 

Coefficient  of  elasticity 3,  208,  209,  512 

Coefficient  of  elasticity,  wrought  iron  in  flexure 516,  517 

Coignet  beton  in  compression 396 

Collapse  of  flues 4 655-659 

Columns,  ends  round,  flat  or  fixed 191 

Columns,  limit  of  application  of  flexure  formulae 195,  196 

Columns,  long,  flexure  of 190 

Columns,  reduction  of  end  sections 479 

Columns,  with  flat  ends 193 

Columns,  with  one  round  and  one  flat  end 194,  195 

Columns,  with  round  ends 194 

Common  column 449 

Common  theory  of  flexure 122 

Common  theory  of  flexure,  unequal  coefficients  of  elasticity 199 

Compression,  greatest  in  bent  beam 204 

Concrete  in  compression 393.  394 


744  INDEX. 


PAGE 

Connections 606 

Continuous  beam 132 

Contra-flexure 131 

Co-ordinates,  equations  in  polar 27 

Co-ordinates,  equations  in  rectangular 14 

Co-ordinates,  equations  in  semi-polar 20 

Copper  in  tension 336,  337,  339,  340,  343-445,  349 

Copper,  tin,  zinc  and  alloys  in  flexure 525,  526 

Copper,  tin,  zinc,  lead  and  alloys  in  compression 386-389 

Copper,  tin,  zinc,  lead  and  alloys  in  shearing 487,  495 

Copper,  tin,  zinc,  lead  and  alloys  in  torsion 487,  507-509 

Crucible  steel,  coefficient  of  elasticity 291 

Crucible  steel,  ultimate  tensile  resistance 294,  303,  324 

Crystallization  of  wrought  iron 259 

Curved  beams,  flexure  of 124-129,  143,  144 

Cylinders,  thick,  hollow 36,  38 

Cylinders,  thin,  hollow 36,  37 


D. 

Deflection  by  common  theory  of  flexure 130 

Deflection  of  cast-iron  flanged  beams 558 

Deflection  of  wrought-iron  T  beams 562,  563 

Diagonal  riveted  joints 642 

Diameter  of  rivets 618,  619,  625,  631,  632,  634,  635,  637,  639 

Distribution  of  stress  in  riveted  joints 607-615 

Drilling  holes  in  steel 325~332 

Driving  and  drawing  spikes 663,  664 

Ductility 211 


E. 

Efficiency  of  riveted  joint 638,  639 

Elastic  limit,  elevation  of 260 

Elastic  limit,  Phoenix  iron  specimens 240 

Elasticity I 

Elasticity,  coefficient  of 3 

Elasticity,  energy  of go 

Elasticity,  limit  of 4?  209 

Elliptical  cylinder,  torsion  of 54 

English  wrought  iron 258 


INDEX.  745 


PAGE 

Eye -bar  heads,  formation  of 645-647,  738 

Eye  bars,  steel,  ultimate  tensile  resistance 297,  738 

Eye  beams,  wrought-iron 567-578 

Euler's  formula • 193 

Euler's  formula  adapted  to  columns 463-468 

Euler's  formula,  limit  of  applicability 477-479 


F. 

Fatigue  of  cast  iron 284,  285 

Fatigue  of  metals 708-722 

Fixed-end  column 191,  193,  433,  437,  439,  441,  442 

Flanged  beams  of  cast  iron 554~559 

Flanged  beams  with  equal  flanges 564-601 

Flanged  beams  with  unequal  flanges 546-564 

Flat-end  column.     See  "  Fixed-end  Column." 

Flexure... 106 

Flexure  by  continuous  normal  load 676,  680 

Flexure  by  oblique  forces 674-679 

Flexure,  coefficients  of  elasticity  for. 512 

Flexure,  coefficients  of  elasticity  unequal 199 

Flexure,  common  theory  of » 122 

Flexure,  formulae  for  rupture 512 

Flexure,  graphical  method 196 

Flexure,  normal  stress  in   112 

Flexure,  theory  of 106 

Flow  of  solids 723-733 

Flues,  collapse  of 655-659 

Flusseisen 263,  266-268 

Fracture  of  wrought  iron 258 


G. 

Glass  in  compression 389 

Glass  in  tension 352,  353 

Gold 351,  352 

Gordon's  formula 409,  430-442 

Grant's  conclusions 362 

Graphical  method  for  flexure 196 

Gun  bronze  in  compression 386 

Gun  bronze  in  tension 342 


746  INDEX. 


PAGE 

Gun  metal  in  tension 341,  343,  346,  347 

Gun  wire,  steel 322 


H. 

Hammering,  effect  on  steel 247,  315-319 

Hardening,  effect  on  steel 3*4 

Hardening,  effect  on  wrought  iron 247 

Hay  steel,  coefficient  of  elasticity 292,  293 

Hay  steel,  ultimate  and  elastic  limits 299,  300 

Hemp  ropes 648-652 

Hodgkinson's  formula , 193,  409,  469-472 

Hooke's  law 2 


I. 

Influence  of  time  on  strains 719,  722 

Intensity  of  continuous  load 136 


K. 

Keene's  cement  in  tension 361 

Keystone  column „ 438,  439,  441 

Kirkaldy's  conclusions 272 


L. 

Laidley's  tests  of  timber  columns 483,  484 

Lanza's  tests  of  timber  columns 481,  482 

Lap  joints 606,  616-632,  639 

Lateral  strains 4 

Latticed  columns 455-460 

Launhardt's  formulae 715,  716 

Lead  in  compression    389 

Lead  in  tension 351,  352 

Length  of  test  piece,  influence  of 230,  231 

Limit  of  elasticity 209 

Load,  intensity  of  continuous 136 

Long  column 371,  409,  430 


INDEX.  747 


PAGE 

Long  column,  flexure  of 190 

Longitudinal  oscillations 100-105 


M. 

Martin  steel  shapes  in  tension 321 

Mill  columns,  timber. 481,  482 

Modulus  of  resilience .- 97 

Modulus  of  rupture  for  flexure  or  bending 129,  515,  545 

Moment,  greatest  bending 137 

Moment,  greatest  or  least  bending 134 

Moments,  bending. 129,  132,  133 

Moment  of  inertia 410,  411 

Moment  of  inertia,  angle  section 417 

Moment  of  inertia,  angle  section,  oblique  axis 429 

Moment  of  inertia,  box  column 413,  414 

Moment  of  inertia,  channel  section,  false 416 

Moment  of  inertia,  channel  section,  true 426 

Moment  of  inertia,  circular  section 423 

Moment  of  inertia,  column  of  plates  and  angles 415 

Moment  of  inertia,  deck  section 427 

Moment  of  inertia,  eye  section,  false 421 

Moment  of  inertia,  eye  section,  true 425 

Moment  of  inertia,  latticed  columns 418,  419 

Moment  of  inertia,  Phoenix  section 424 

Moment  of  inertia,  rectangular  sections 422,  423 

Moment  of  inertia,  star  section 421 

Moment  of  inertia,  tee  section 420 

Mortise  holes,  shearing  behind 664,  665 

Muntz  metal  in  tension 343,  347 


N. 

Neutral  axis 114 

Neutral  curve,  cantilever 171 

Neutral  curve,  continuous  beam 177 

Neutral  curve,  non-continuous  beam 174 

Ne'Ural  curve,  special  cases 171 

Neutral  surface 114 

Non-continuous  beam 132 


748  INDEX. 


o. 


PAGE 


Open-hearth  steel  in  tension 297,  306,  311,  326,  327 

Oscillations,  longitudinal 100-105 

Oscillations,  torsional - 78 

Overlap  in  riveted  joint 615,  626,  627,  635 


P. 

Palladium 351,  352 

Parian  cement  in  tension 354,  361 

Permanent  set •. 211 

Phoenix  columns 438,  439,  442,  448-455 

Phosphor  bronze 344,  347 

Pin  connection 644-648 

Pin  end  columns • 437,  439 

Pitch  of  rivets 616,  617,  621,  623,  626,  629,  632-634,  637,  639 

Plates,  bulging  of 665,  674 

Platinum 351,  352 

Portland  cement  and  mortar  in  tension 354,  357,  359,  360 

Potential  energy  of  elasticity 94,  96 

Practical  formulae  for  solid  beams 543,  544 

Pressure  on  rivets 613,  619-624,  627-629,  632,  634-637 

Punching,  effect  of  in  riveted  joints 615,  624,  634 

Punching  steel,  effect  of 325-332 


R. 

Rail  steel 295 

Reactions  under  continuous  beams 157,  171 

Reaming  holes  in  steel 325-332 

Rectangular  cylinder,  torsion  of .' 59 

Red  brass  in  tension 350 

Reduction  of  column  ends 479 

Reduction  of  resistance  between  ultimate  and  breaking  points 269 

Resilience 96,  97 

Resilience,  modulus  of '97 

Rivet  steel 315 

Riveted  joints 606-643 

Riveted  joints,  friction  of 642 


INDEX.  749 


PAGE 

Riveted  truss  joints 640-642,  737 

Rollers,  resistance  of 659-662,  693,  694 

Rolling,  effect  of,  on  steel 315-319 

Ropes,  iron,  steel  and  hemp 648-652 

Round  end  columns 191,  194,  433,  439,  442 


s. 

Safety  factor 681 

Sandberg's  conclusions  regarding  low  temperatures 253 

Shape  iron  in  tension 257 

Shape  steel  in  tension 321 

Shapes,  tensile  resistance  of 238 

Shear,  counter 137 

Shear,  greatest  bending  intensity  in  rectangular  beam 121 

Shear,  greatest  total  in  non-continuous  beam 137 

Shear,  in  bent  beam 133 

Shear,  main 137 

Shearing  behind  mortise  holes 664,  665 

Shearing,  coefficient  of  elasticity 4,  6,  487-490 

Shearing,  greatest  in  bent  beam 206 

Shearing  of  rivets 621,  624,  625,  630-632,  637 

Shearing,  ultimate  resistance 490-498 

Short  blocks 371 

Siemens-Martin  steel 324 

Siemen's  steel 312 

Size  of  test  piece,  effect  of 212,  224,  225 

Skin  of  bar,  resistance  of 233 

Solid  beams,  rectangular  and  circular 514,  544 

Spangenberg's  experiments 709 

Specifications,  Franklin  Square  bridge 694-698 

Specifications,  Greenbush  bridge 686-688 

Specifications,  Menomonee  bridge 689-694 

Specifications,  Niagara  suspension  bridge 688 

Specifications,  Plattsmouth  bridge 701-703 

Specifications,  railway  bridges 699,  700 

Specifications,  Sabula  bridge 682-686 

Specifications,  steel  cable  wire,  East  River  bridge 703-705 

Specifications,  steel  wire  rope,  East  River  bridge 705,  706 

Specifications,  steel  work  for  East  River  bridge 706,  707 

Spheres,  thick,  hollow 84 

Spikes,  driving  and  drawing 663,  664 


750  INDEX. 


PAGE 

Square  columns 438,  439>  441,  448 

Steel  columns 44§ 

Steel  in  compression 380 

Coefficient  of  elasticity 380,  382,  383 

Elastic  limit  and  ultimate  resistance 380-385 

Effect  of  tempering 381 

Effect  of  annealing 382 

(For  various  grades  and  varieties  see  text  under  preceding  heads.) 

Steel  in  shearing 487,  493-495 

Steel  in  tension 286 

Coefficient  of  elasticity 286 

Ultimate  resistance  and  elastic  limit 294 

Boiler  plate 306 

Hardening  and  tempering  steel  plate 314 

Rivet  steel 315 

Reduction  of  section  by  hammering  and  rolling 315 

Annealing  steel 319 

Steel  wire 319 

Shape  steel 321 

Gun  wire 322 

Effect  of  low  and  high  temperature  on  steel 323 

Constructive  manipulations,  such  as  punching,  drilling  and  reaming.   325 

Bauschinger's  experiments 333 

Fracture  of  steel 334 

Effect  of  chemical  composition 334 

Steel  in  torsion 487,  504-507 

Steel  plate,  coefficient  of  elasticity 290,  291,  292 

Steel  plate,  ultimate  tensile  resistance 294,  306,  314,  326,  327 

Sterro-metal  in  tension 343 

Stones,  natural  building  in  compression 398-402 

Strain I 

Strains,  influence  of  time  on 719-722 

Strains,  lateral 4 

Stress  I 

Stresses,  expressions  for  tangential  and  direct 8 

Stresses,  greatest  at  any  point  in  a  beam 2O2 

Stresses,  plane  of  greatest  normal  in  a  beam \ 204 

Stresses,  plane  of  greatest  shearing  in  a  beam 206 

Styffe's  conclusions  regarding  low  temperature 252 

Suddenly  applied  forces  or  loads 98,  100 

Suddenly  applied  stress,  resistance  of  iron  to 269 

Swelled  columns 434,  439,  441 


INDEX.  75 1 


T. 

PACE 

Temperature,  effect  of  increase  on  wrought  iron 247-251,  324,  347 

Temperature,  effect  of  low  on  wrought  iron 251-254 

Temperatures,  effect  of  high  and  low  on  steel 323,  324 

Tempering,  effect  on  steel 314 

Tension,  greatest  in  bent  beam 204 

Theorem  of  three  moments 146 

Theorem  of  three  moments,  common  form 154,  158 

Theory  of  flexure,  general  formulae 143 

Thick,  hollow  cylinders 36,  38 

Thin,  hollow  cylinders 36,  37 

Thick,  hollow  spheres 84 

Thickness  of  web  plate  in  flanged  beams. 601-603 

Thurston's  conclusions  regarding  low  temperatures 253 

Thurston's  relation  between  tension  and  torsion 510 

Timber  beams 526-536 

Timber  beams  of  natural  and  prepared  woods 536 

Timber  columns,  C.  Shaler  Smith's  formulae 485,  486 

Timber  columns  or  pillars 471,  480-486 

Timber  in  compression 403-408 

Timber  in  shearing ; 487,  488,  496-498 

Timber  in  tension 365-370 

Timber  in  torsion   487,  488,  509,  5 10 

Tin  in  compression 387 

Tin  in  tension 336,  337,  339,  343 

Tobin's  alloy  in  tension 336,  338,  339,  341 

Tobin's  alloy  in  flexure 525,  526 

Torsion,  coefficient  of  elasticity 8,  487-490,  498 

Torsion,  general  observations 77 

Torsion,  greatest  shear  in  circular  sections 76 

Torsion,  greatest  shear  in  elliptical  sections 56 

Torsion,  greatest  shear  in  rectangular  sections 71,  74,  75 

Torsion,  greatest  shear  in  triangular  sections 59 

Torsion  in  equilibrium 43 

Torsion,  moment  of  circular  sections 76 

Torsion,  moment  of  elliptical  sections 55,  56 

Torsion,  moment  of  rectangular  sections 70,  74,  75 

Torsion,  moment  of  triangular  sections 58,  59 

Torsion  of  circular  section 75 

Torsion  of  elliptical  section 54 

Torsion  of  rectangular  section 59 

Torsion  of  triangular  section 57 


752  '  INDEX. 


PAGE 


Torsion  pendulum 82,  83 

Torsional  oscillations 78 

Townsend's  experiments,  flow  of  solids 725 

Tresca's  experiments,  flow  of  solids 724 

Tresca's  hypotheses,  flow  of  solids 725~727 

Triangular  cylinders,  torsion  of 57 

Tubes  as  columns 473>  475 

Tubes,  collapse  of 655-659 


u. 

Ultimate  resistance 210 

Ultimate  resistance,  elevation  of 260 

Ultimate  resistance  of  wrought  iron  along  or  across  fibres 242,  243 

Ultimate  resistance,  Phoenix  iron  specimens 240 

Unequal  coefficients  of  elasticity,  flexure  with 199 


w. 

Web  plate  of  flanged  beam,  thickness  of 601,  603 

Welded  joints 643,  644 

Weyrauch's  formula 717,  718 

Whitworth's  compressed  steel 305 

Wire,  brass 341,  344 

Wire,  copper 341,  344 

Wire,  Fairbairn's  tests 257 

Wire,  phosphor  bronze     344 

Wire,  Roebling's  tests  on  wrought  iron 254 

Wire  ropes 648-652 

Wire,  steel 319,  320 

Wire,  steel  gun 322 

Wire,  Thurston's  tests  on  old  wrought  iron 255 

Woehler's  experiments 710-714 

Woehler's  law 708 

Working  stress 681 

Wrought-iron  chain  cables 652-654 

Wrought-iron  columns,  solid  rectangular 447,  474,  475 

Wrought-iron  columns,  solid  round 447,  470 

Wrought  iron  in  compression 372 

Coefficient  of  elasticity 373,  374 

Elastic  limit  and  ultimate  resistance 374~37° 


INDEX.  753 


PACE 

Wrought  iron  in  shearing 487,  491-493 

Wrought  iron  in  tension 212 

Coefficient  of  elasticity 212-223 

Effect  of  size 212 

Rounds  and  flats 213 

Plates  by  Franklin  Institute  committee 215 

St.  Louis  bridge  specimens 216 

Values  of  coefficient  to  ultimate  resistance 217 

Graphical  representation ...   220 

Ultimate  resistance  and  elastic  limit.  . 223 

Influence  of  size  and  dimensions 224-226 

Values  for  large  bars 227 

Reduction  of  piles 228,  229 

Influence  of  length 230,  231 

Influence  of  skin  of  bar. 233 

Shapes 233 

Large  bars  and  rounds 234,  235 

Specimens  from  bars,  plates  and  angles 237-239 

Boiler  plate 24 1 

Effect  of  annealing  245 

Effect  of  hardening 247 

Variation  of  resistance  with  increase  of  temperature 247 

Effect  of  low  temperature 251 

Iron  wire 254 

Resistance  of  shape  iron 257 

English  wrought  iron 258 

Fracture  of  wrought  iron . 258 

Crystallization  of  wrought  iron 259 

Elevation  of  ultimate  resistance  and  elastic  limit 260 

Bauschinger's  experiments  on  the  change  of  elastic  limit  and  coeffi- 
cient of  elasticity ...    262 

Resistance  of  bar  iron  to  suddenly  applied  stress 269 

Reduction  of  resistance  between  the  ultimate  and  breaking  point. . .   269 

Effect  of  chemical  constitution 270 

Kirkalcly's  conclusions 272 

Wrought  iron  in  torsion 487,  498-500 

Wrought  iron  in  tee  beams 559~5°3 


z. 

Zinc  in  compression 388 

Zinc  in  tension 336»  339»  343 

48 


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